STAT,
"VRT
A COURSE IN
BY
EDOUARD GOURSAT
PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF PARIS
TRANSLATED BY
EARLE RAYMOND HEDRICK
PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF MISSOURI
VOL. I
DERIVATIVES AND DIFFERENTIALS
DEFINITE INTEGRALS EXPANSION IN SERIES
APPLICATIONS TO GEOMETRY
GINN AND COMPANY
BOSTON • NEW YORK • CHICAGO • LONDON
ATLANTA • DALLAS • COLUMBUS • SAN FKANCISCO
STAT.
LIBRARY
ENTERED AT STATIONERS1 HALL
COPYRIGHT, 1904, BY
EARLE RAYMOND HEDRICK
ALL RIGHTS RESERVED
PRINTED IN THE UNITED STATES OF AMERICA
426.6
jgregg
GINN AND COMPANY • PRO
PRIETORS • BOSTON • U.S.A.
AUTHOR'S PREFACE
This book contains, with slight variations, the material given in
my course at the University of Paris. I have modified somewhat
the order followed in the lectures for the sake of uniting in a single
volume all that has to do with functions of real variables, except
the theory of differential equations. The differential notation not
being treated in the " Classe de Mathematiques speciales," * I have
treated this notation from the beginning, and have presupposed only
a knowledge of the formal rules for calculating derivatives.
Since mathematical analysis is essentially the science of the con
tinuum, it would seem that every course in analysis should begin,
logically, with the study of irrational numbers. I have supposed,
however, that the student is already familiar with that subject. The
theory of incommensurable numbers is treated in so many excellent
well-known works f that I have thought it useless to enter upon such
a discussion. As for the other fundamental notions which lie at the
basis of analysis, — such as the upper limit, the definite integral, the
double integral, etc., — I have endeavored to treat them with all
desirable rigor, seeking to retain the elementary character of the
work, and to avoid generalizations which would be superfluous in a
book intended for purposes of instruction.
Certain paragraphs which are printed in smaller type than the
body of the book contain either problems solved in detail or else
*An interesting account of French methods of instruction in mathematics will
be found in an article by Pierpont, Bulletin Amer. Math. Society, Vol. VI, 2d series
(1900), p. 225.— TRANS.
t Such books are not common in English. The reader is referred to Pierpont,
Theory of Functions of Real Variables, Ginn & Company, Boston, 1905; Tannery,
Lemons d'arithiiietique, 1900, and other foreign works on arithmetic and on real
functions.
iii
7814G2
iv AUTHOR'S PREFACE
supplementary matter which the reader may omit at the first read
ing without inconvenience. Each chapter is followed by a list of
examples which are directly illustrative of the methods treated in
the chapter. Most of these examples have been set in examina
tions. Certain others, which are designated by an asterisk, are
somewhat more difficult. The latter are taken, for the most part,
from original memoirs to which references are made.
Two of my old students at the Ecole Normale, M. Emile Cotton
and M. Jean Clairin, have kindly assisted in the correction of proofs ;
I take this occasion to tender them my hearty thanks.
E. GOURSAT
JANUARY 27, 1902
TRANSLATOR'S PREFACE
The translation of this Course was undertaken at the suggestion
of Professor W. F. Osgood, whose review of the original appeared
in the July number of the Bulletin of the American Mathematical
Society in 1903. The lack of standard texts on mathematical sub
jects in the English language is too well known to require insistence.
I earnestly hope that this book will help to fill the need so generally
felt throughout the American mathematical world. It may be used
conveniently in our system of instruction as a text for a second course
in calculus, and as a book of reference it will be found valuable to
an American student throughout his work.
Few alterations have been made from the French text. Slight
changes of notation have been introduced occasionally for conven
ience, and several changes and .additions have been made at the sug
gestion of Professor Goursat, who has very kindly interested himself
in the work of translation. To him is due all the additional matter
not to be found in the French text, except the footnotes which are
signed, and even these, though not of his initiative, were always
edited by him. I take this opportunity to express my gratitude to
the author for the permission to translate the work and for the
sympathetic attitude which he has consistently assumed. I am also
indebted to Professor Osgood for counsel as the work progressed
and for aid in doubtful matters pertaining to the translation.
The publishers, Messrs. Ginn & Company, have spared no pains to
make the typography excellent. Their spirit has been far from com
mercial in the whole enterprise, and it is their hope, as it is mine,
that the publication of this book will contribute to the advance of
mathematics in America. E R HEDRICK
AUGUST, 1904
CONTENTS
CHAPTER PAGE
I. DERIVATIVES AND DIFFERENTIALS 1
I. Functions of a Single Variable 1
II. Functions of Several Variables 11
III. The Differential Notation 19
II. IMPLICIT FUNCTIONS. FUNCTIONAL DETERMINANTS. CHANGE
OF VARIABLE 35
I. Implicit Functions ........ 35
II. Functional Determinants ...... 52
III. Transformations ... .... 61
III. TAYLOR'S SERIES. ELEMENTARY APPLICATIONS. MAXIMA
AND MINIMA ........ 89
I. Taylor's Series with a Remainder. Taylor's Series . 89
II. Singular Points. Maxima and Minima . . . .110
IV. DEFINITE INTEGRALS ........ 134
I. Special Methods of Quadrature . . . . .134
II. Definite Integrals. Allied Geometrical Concepts . . 140
III. Change of Variable. Integration by Parts . . .166
IV. Generalizations of the Idea of an Integral. Improper
Integrals. Line Integrals ...... 175
V. Functions defined by Definite Integrals .... 192
VI. Approximate Evaluation of Definite Integrals . .196
V. INDEFINITE INTEGRALS 208
I. Integration of Rational Functions ..... 208
II. Elliptic and Hyperelliptic Integrals .... 226
III. Integration of Transcendental Functions . . .236
VI. DOUBLE INTEGRALS ........ 250
I. Double Integrals. Methods of Evaluation. Green's
Theorem 250
II. Change of Variables. Area of a Surface . . . 264
III. Generalizations of Double Integrals. Improper Integrals.
Surface Integrals ....... 277
IV. Analytical and Geometrical Applications . . . 284
viii CONTENTS
CHAPTER PAGE
VII. MULTIPLE INTEGRALS. INTEGRATION OF TOTAL DIFFER
ENTIALS 296
I. Multiple Integrals. Change of Variables . . . 296
II. Integration of Total Differentials . . . . .313
VIII. INFINITE SERIES . •. 327
I. Series of Real Constant Terms. General Properties.
Tests for Convergence 327
II. Series of Complex Terms. Multiple Series . . . 350
III. Series of Variable Terms. Uniform Convergence . . 360
IX. POWER SERIES. TRIGONOMETRIC SERIES .... 375
I. Power Series of a Single Variable . . . . . 375
II. Power Series in Several Variables ..... S94
III. Implicit Functions. Analytic Curves and Surfaces . 399
IV. Trigonometric Series. Miscellaneous Series . . .411
X. PLANE CURVES 426
I. Envelopes 426
II. Curvature 433
III. Contact of Plane Curves 443
XI. SKEW CURVES 453
I. Osculating Plane ........ 453
II. Envelopes of Surfaces . . • . . . . . 459
III. Curvature and Torsion of Skew Curves .... 468
IV. Contact between Skew Curves. Contact between Curves
and Surfaces ........ 486
XII. SURFACES 497
I. Curvature of Curves drawn on a Surface . . . 497
II. Asymptotic Lines. Conjugate Lines .... 506
III. Lines of Curvature . . . . . . . .514
IV. Families of Straight Lines 526
INDEX . 541
CHAPTER I
DERIVATIVES AND DIFFERENTIALS
I. FUNCTIONS OF A SINGLE VARIABLE
1. Limits. When the successive values of a variable x approach
nearer and nearer a constant quantity a, in such a way that the
absolute value of the difference x — a finally becomes and remains
less than any preassigned number, the constant a is called the
limit of the variable x. This definition furnishes a criterion for
determining whether a is the limit of the variable x. The neces
sary and sufficient condition that it should be, is that, given any
positive number e, no matter how small, the absolute value of x — a
should remain less than e for all values which the variable x can
assume, after a certain instant.
Numerous examples of limits are to be found in Geometry
and Algebra. For example, the limit of the variable quantity
x = (a2 — m2) / (a — m), as m approaches a, is 2 a ; for x — 2 a will
be less than e whenever m — a is taken less than e. Likewise, the
variable x = a — 1/n, where n is a positive integer, approaches the
limit a when n increases indefinitely ; for a — x is less than e when
ever n is greater than 1/e. It is apparent from these examples that
the successive values of the variable x, as it approaches its limit, may
form a continuous or a discontinuous sequence.
It is in general very difficult to determine the limit of a variable
quantity. The following proposition, which we will assume as self-
evident, enables us, in many cases, to establish the existence of a limit.
Any variable quantity which never decreases, and which ahvays
remains less than a constant quantity L, approaches a limit I, which
is less than or at most equal to L.
Similarly, any variable quantity which never increases, and which
always remains greater than a constant quantity L', approaches a
limit l'} which is greater than or else equal to L'.
1
2. DERIVATIVES AND DIFFERENTIALS [I, §2
For example, if each of an infinite series of positive terms is
less, respectively, than the corresponding term of another infinite
series of positive terms which is known to converge, then the first
series converges also ; for the sum 2n of the first n terms evidently
increases with n, and this sum is constantly less than the total sum
5 of the second series.
2. Functions. When two variable quantities are so related that
the value of one of them depends upon the value of the other, they
are said to be functions of each other. If one of them be sup
posed to vary arbitrarily, it is called the independent variable. Let
this variable be denoted by x, and let us suppose, for example,
that it can assume all values between two given numbers a and b
(a < b). Let y be another variable, such that to each value of x
between a and b, and also for the values a and b themselves, there
corresponds one definitely determined value of y. Then y is called
a function of x, defined in the interval (a, b) ; and this dependence
is indicated by writing the equation y =/(z). For instance, it may
happen that y is the result of certain arithmetical operations per
formed upon x. Such is the case for the very simplest functions
studied in elementary mathematics, e.g. polynomials, rational func
tions, radicals, etc.
A function may also be defined graphically. Let two coordinate
axes Ox, Oy be taken in a plane ; and let us join any two points A
and B of this plane by a curvilinear arc .4 CB, of any shape, which
is not cut in more than one point by any parallel to the axis Oy.
Then the ordinate of a point of this curve will be a function of the
abscissa. The arc A CB may be composed of several distinct por
tions which belong to different curves, such as segments of straight
lines, arcs of circles, etc.
In short, any absolutely arbitrary law may be assumed for finding
the value of y from that of x. The word function, in its most gen
eral sense, means nothing more nor less than this : to every value of
x corresponds a value of y.
3. Continuity. The definition of functions to which the infini
tesimal calculus applies does not admit of such broad generality.
Let y =f(x) be a function defined in a certain interval (a, b), and
let x0 and x0 -f h be two values of x in that interval. If the differ
ence f(x0 -f A) — f(xo) approaches zero as the absolute value of h
approaches zero, the function f(x} is said to be continuous for the
value x0. From the very definition of a limit we may also say that
I, §3] FUNCTIONS OF A SINGLE VARIABLE 3
a function f(x) is continuous for x — x0 if, corresponding to every
positive number e, no matter how small, we can find a positive num
ber 77, such that
|/(*o + A)-/(*o)|<«
for every value of h less than rj in absolute value.* We shall say that
a function f(x) is continuous in an interval (a, b) if it is continuous
for every value of x lying in that interval, and if the differences
each approach zero when h, which is now to be taken only positive,
approaches zero.
In elementary text-books it is usually shown that polynomials,
rational functions, the exponential and the logarithmic function,
the trigonometric functions, and the inverse trigonometric functions
are continuous functions, except for certain particular values of
the variable. It follows directly from the definition of continuity
that the sum or the product of any number of continuous functions
is itself a continuous function ; and this holds for the quotient of
two continuous functions also, except for the values of the variable
for which the denominator vanishes.
It seems superfluous to explain here the reasons which lead us to
assume that functions which are defined by physical conditions are,
at least in general, continuous.
Among the properties of continuous functions we shall now state
only the two following, which one might be tempted to think were
self-evident, but which really amount to actual theorems, of which
rigorous demonstrations will be given later, f
I. If the function y—f(x) is continuous in the interval (a, b), and
if N is a number between f (a) andf(b), then the equation f(x) = N
has at least one root between a and b.
II. There exists at least one value of x belonging to the interval
(a, b'), inclusive of its end points, for which y takes on a value M
which is greater than, or at least equal to, the value of the function at
any other point in the interval. Likewise, there exists a value of x
for which y takes on a value m, than which the function assumes no
smaller value in the interval.
The numbers M and m are called the maximum and the minimum
values of f(x), respectively, in the interval (a, b*). It is clear that
* The notation | a \ denotes the absolute value of a.
t See Chapter IV.
4 DERIVATIVES AND DIFFERENTIALS [I, §4
the value of x for which /(ce) assumes its maximum value M, or the
value of x corresponding to the minimum m, may be at one of the
end points, a or b. It follows at once from the two theorems above,
that if N is a number between M and m, the equation /(«) = N has
at least one root which lies between a and b.
4. Examples of discontinuities. The functions which we shall study
will be in general continuous, but they may cease to be so for
certain exceptional values of the variable. We proceed to give
several examples of the kinds of discontinuity which occur most
frequently.
The function y = 1 / (x — a) is continuous for every value x0 of
x except a. The operation necessary to determine the value of y
from that of x ceases to have a meaning when x is assigned the
value a ; but we note that when x is very near to a the absolute
value of y is very large, and y is positive or negative with x — a.
As the difference x — a diminishes, the absolute value of y increases
indefinitely, so as eventually to become and remain greater than any
preassigned number. This phenomenon is described by saying that
y becomes infinite when x = a. Discontinuity of this kind is of
great importance in Analysis.
Let us consider next the function y = sin 1/z. As x approaches
zero, I/a; increases indefinitely, and y does not approach any limit
whatever, although it remains between + 1 and — 1. The equation
sin l/a; = ,4, where A \ < 1, has an infinite number of solutions
which lie between 0 and e, no matter how small e be taken. What
ever value be assigned to y when x — 0, the function under con
sideration cannot be made continuous for x = 0.
An example of a still different kind of discontinuity is given by
the convergent infinite series
When x approaches zero, S (x~) approaches the limit 1, although
5 (0) = 0. For, when x = 0, every term of the series is zero, and
hence 5 (0) = 0. But if x be given a value different from zero, a
geometric progression is obtained, of which the ratio is 1/(1 + a;2).
Hence
~
I, §5] FUNCTIONS OF A SINGLE VARIABLE 5
and the limit of S(x) is seen to be 1. Thus, in this example, the
function approaches a definite limit as x approaches zero, but that
limit is different from the value of the function for x = 0.
5. Derivatives. Let/(x) be a continuous function. Then the two
terms of the quotient
k
approach zero simultaneously, as the absolute value of h approaches
zero, while x remains fixed. If this quotient approaches a limit,
this limit is called the derivative of the function /(#), and is denoted
by y', or by /' (x), in the notation due to Lagrange.
An important geometrical concept is associated with this analytic
notion of derivative. Let us consider, in a plane XOY, the curve
A MB, which represents the function y =/(#), which we shall assume
to be continuous in the interval (a, b). Let M and M' be two points
on this curve, in the interval (a, b), and let their abscissas be x and
x + A, respectively. The slope of the straight line MM' is then
precisely the quotient above. Now as h approaches zero the point
M ' approaches the point M] and, if the function has a derivative,
the slope of the line MM' approaches the limit y'. The straight line
MM', therefore, approaches a limiting position, which is called the
tangent to the curve. It follows that the equation of the tangent is
Y-y = y'(X-x),
where X and Y are the running coordinates.
To generalize, let us consider any curve in space, and let
be the coordinates of a point on the curve, expressed as functions of
a variable parameter t. Let M and M' be two points of the curve
corresponding to two values, t and t + h, of the parameter. The
equations of the chord MM1 are then
x-f(t) Y
f(t + h) -
If we divide each denominator by h and then let h approach zero,
the chord MM' evidently approaches a limiting position, which is
given by the equations
X -f(f) Y-
f(t) 4,' ft)
6 DERIVATIVES AND DIFFERENTIALS [i,§5
provided, of course, that each of the three functions f(t), <f> (t), \J/ (t)
possesses a derivative. The determination of the tangent to a curve
thus reduces, analytically, to the calculation of derivatives.
Every function which possesses a derivative is necessarily con
tinuous, but the converse is not true. It is easy to give examples
of continuous functions which do not possess derivatives for par
ticular values of the variable. The function y = xsinl/x, for
example, is a perfectly continuous function of x, for x = 0,* and y
approaches zero as x approaches zero. But the ratio y /x = sinl/cc
does not approach any limit whatever, as we have already seen.
Let us next consider the function y = x*. Here y is continuous
for every value of a;; and y = 0 when x = 0. But the ratio y /x = x~*
increases indefinitely as x approaches zero. For abbreviation the
derivative is said to be infinite for x = 0 ; the curve which repre
sents the function is tangent to the axis of y at the origin.
Finally, the function
y =
is continuous at x = 0,* but the ratio y /x approaches two different
limits according as x is always positive or always negative while
it is approaching zero. When x is positive and small, el/x is posi
tive and very large, and the ratio y /x approaches 1. But if x
is negative and very small in absolute value, el/x is very small, and
the ratio y / x approaches zero. There exist then two values of the
derivative according to the manner in which x approaches zero : the
curve which represents this function has a corner at the origin.
It is clear from these examples that there exist continuous func
tions which do not possess derivatives for particular values of the
variable. But the discoverers of the infinitesimal calculus confi
dently believed that a continuous function had a derivative in gen
eral. Attempts at proof were even made, but these were, of course,
fallacious. Finally, Weierstrass succeeded in settling the question
conclusively by giving examples of continuous functions which do not
possess derivatives for any values of the variable whatever.! But
as these functions have not as yet been employed in any applications,
* After the value zero has been assigned to y for x = 0. — TRANSLATOR.
t Note read at the Academy of Sciences of Berlin, July 18, 1872. Other examples
are to be found in the memoir by Darboux on discontinuous functions (Annales de
I'Ecole Normale Superieure, Vol. IV, 2d series). One of Weierstrass's examples is
given later (Chapter IX).
I, §6] FUNCTIONS OF A SINGLE VARIABLE 7
we shall not consider them here. In the future, when we say that
a function f(x) has a derivative in the interval (a, b), we shall mean
that it has an unique finite derivative for every value of x between
a and b and also f or x = a (h being positive) and f or x = b (h being
negative), unless an explicit statement is made to the contrary.
6. Successive derivatives. The derivative of a function f(x) is in
general another function of x,f'(x). If f'(x) in turn has a deriva
tive, the new function is called the second derivative of /(x), and is
represented by y" or by f"(x). In the same way the third deriva
tive y'", or /'"(#), is defined to be the derivative of the second, and
so on. In general, the rath derivative 7/n), or fw(x), is the deriva
tive of the derivative of order (n — 1). If, in thus forming the
successive derivatives, we never obtain a function which has no
derivative, we may imagine the process carried on indefinitely. In
this way we obtain an unlimited sequence of derivatives of the func
tion /(cc) with which we started. Such is the case for all functions
which have found any considerable application up to the present
time.
The above notation is due to Lagrange. The notation Dny, or
Dnf(x), due to Cauchy, is also used occasionally to represent the
wth derivative. Leibniz' notation will be given presently.
7. Rolle's theorem. The use of derivatives in the study of equa
tions depends upon the following proposition, which is known as
Roue's Theorem :
Let a and b be two roots of the equation f (x) = 0. If the function
f(x) is continuous and possesses a derivative in the interval (a, b~),
the equation /'(#) = 0 has at least one root which lies between a and b.
For the function f(x) vanishes, by hypothesis, for x = a and x = b.
If it vanishes at every point of the interval (a, b), its derivative also
vanishes at every point of the interval, and the theorem is evidently
fulfilled. If the function f(x) does not vanish throughout the inter
val, it will assume either positive or negative values at some points.
Suppose, for instance, that it has positive values. Then it will have
a maximum value M for some value of x, say xlf which lies between
a and b (§ 3, Theorem II). The ratio
8 DERIVATIVES AND DIFFERENTIALS [I, §8
where h is taken positive, is necessarily negative or else zero.
Hence the limit of this ratio, i.e. f'(x^), cannot be positive ; i.e.
f'(xi) = 0- But if we consider f'(x\) as the limit of the ratio
>
— h
where h is positive, it follows in the same manner that f\x\) ^ 0,
From these two results it is evident that/'^) = 0.
8. Law of the mean. It is now easy to deduce from the above
theorem the important law of the mean : *
Let f(x) be a continuous function which has a derivative in the
interval (a, b). Then
(1) m-f(a) = (b-a)f(c-),
where c is a number between a and b.
In order to prove this formula, let <£ (x) be another function which
has the same properties as/(x), i.e. it is continuous and possesses a
derivative in the interval (a, b). Let us determine three constants,
A, B, C, such that the auxiliary function
vanishes for x = a and for x = b. The necessary and sufficient
conditions for this are
A /(a) +B <£(«)+ C' = 0, Af(b) + B<l>(b)+ C = 0;
and these are satisfied if we set
A = <l>(a)-4> (b), B =/(&) -/(a), C =/(«) 0 (*)-/(&) * (a).
The new function \J/(x) thus defined is continuous and has a derivative
in the interval (a, b). The derivative if/'(x) = A f'(x) + B <£'(z) there
fore vanishes for some value c which lies between a and b, whence
replacing A and B by their values, we find a relation of the form
It is merely necessary to take <£ (a;) = x in order to obtain the equality
which was to be proved. It is to be noticed that this demonstration
does not presuppose the continuity of the derivative/'^).
•"Formule des accroissements finis." The French also use " Formule de la
moyenne" as a synonym. Other English synonyms are "Average value theorem "
and " Mean value theorem." —TRANS.
I, §8] FUNCTIONS OF A SINGLE VARIABLE 9
From the theorem just proven it follows that if the derivative
f'(x) is zero at each point of the interval (a, b), the function f(x)
has the same value at every point of the interval ; for the applica
tion of the formula to two values Xi, xz, belonging to the interval
(a, b), gives f(xi)=f(x.2). Hence, if two functions have the same
derivative, their difference is a constant ; and the converse is evi
dently true also. If a function F(x) be given whose derivative is
f(oc), all other functions which have the same derivative are found by
adding to F(x) an arbitrary constant*
The geometrical interpretation of the equation (1) is very simple,
Let us draw the curve A MB which represents the function y = f(x)
in the interval (a, b). Then the ratio [/(&)— /(«)]/ (b — a) is the
slope of the chord AB, while /'(«) is the slope of the tangent at a
point C of the curve whose abscissa is c. Hence the equation (1)
expresses the fact that there exists a point C on the curve A MB,
between A and B, where the tangent is parallel to the chord AB.
If the derivative /'(a;) is continuous, and if we let a and b approach
the same limit x0 according to any law whatever, the number c,
which lies between a and b, also approaches x0} and the equation (1)
shows that the limit of the ratio
b — a
is f'(xo). The geometrical interpretation is as follows. Let us
consider upon the curve y=f(x) a point M whose abscissa is x0,
and two points A and B whose abscissa are a and b, respectively.
The ratio [/(&) —/(«)] / (b — «) is equal to the slope of the chord
AB, while /'(x0) is the slope of the tangent at M. Hence, when
the two points A and B approach the point M according to any law
whatever, the secant AB approaches, as its limiting position, the
tangent at the point M.
* This theorem is sometimes applied without due regard to the conditions imposed in
its statement. Let/(x) and 0(^), f°r example, be two continuous functions which have
derivatives /'(a;), </)'(x) in an interval (a, 6). If the relation /'(z) <t>(x)—f(x) 4>'(x) = 0
is satisfied by these four functions, it is sometimes accepted as proved that the deriva
tive of the function// <f>, or [/'(a;) 0 (cc) - f(x) 0'(z)] / <£2, is zero, and that accordingly
f/<t> is constant in the interval (a, b). But this conclusion is not absolutely rigorous
unless the function $ (a;) does not vanish in the interval (a, b). Suppose, for instance,
that 0 (a;) and <j>'(x) both vanish for a value c between a and 6. A function/(x) equal
to Ci<f>(x) between a and c, and to C%<f)(x) between c and b, where Cj and C2 are dif
ferent constants, is continuous and has a derivative in the interval (a, b), and we have
f'(x)<t>(x) — f(x)<p'(x) = 0 for every value of x in the interval. The geometrical
interpretation is apparent.
10
DERIVATIVES AND DIFFERENTIALS
[I, §9
This does not hold in general, however, if the derivative is not
continuous. For instance, if two points be taken on the curve
y = x*, on opposite sides of the y axis, it is evident from a figure
that the direction of the secant joining them can be made to approach
any arbitrarily assigned limiting value by causing the two points to
approach the origin according to a suitably chosen law.
The equation (!') is sometimes called the generalized law of the
mean. From it de 1' Hospital's theorem on indeterminate forms fol
lows at once. For, suppose f(a) = 0 and <f> (a) = 0. Replacing b
by x in (!'). we find
* \ /• \
where a^ lies between a and x. This equation shows that if the
ratio f'(x)/(j>'(x) approaches a limit as x approaches a, the ratic
/"(#) / (j> (a;) approaches the same limit, if f(a) = 0 and <f> (a) = 0.
9. Generalizations of the law of the mean. Various generalizations of the law
of the mean have been suggested. The following one is due to Stieltjes (Bulletin
de la Socie'te Mathtmatique, Vol. XVI, p. 100). For the sake of defmiteness con
sider three functions, /(x), g(x), h(x), each of which has derivatives of the first
and second orders. Let a, 6, c be three particular values of the variable (a < b < c).
Let A be a number defined by the equation
and let
be an auxiliary function. Since this function vanishes when x = b and when
x = c, its derivative must vanish for some value f between 6 and c. Hence
/(«)
9 (a)
h(a)
1
a
a«
/(&)
9(b)
h(b)
-A
1
b
62
/(c)
9(c]
h(c)
1
c
c2
/(a)
9(0.}
h(a)
1
a
a2
/(&)
9(b)
h(b)
-A
1
b
b2
/(*)
9(x)
h(x)
1
x
x*
/(a) g(a) h(a)
/(&) g(b) h(b)
/'(f)
-A
1 a a2
1 b b2
0 1 2f
If b be replaced by x in the left-hand side of this equation, we obtain a function
of x which vanishes when x = a and when x = b. Its derivative therefore van
ishes for some value of x between a and 6, which we shall call £. The new
equation thus obtained is
/ (a) g (a) h (a)
/'(f)
1 2f
= 0.
Finally, replacing f by x in the left-hand side of this equation, we obtain a func
tion of x which vanishes when x =• £ and when x •= f . Its derivative vanishes
I, §10] FUNCTIONS OF SEVERAL VARIABLES 11
for some value ij, which lies between £ and f and therefore between a and c.
Hence A must have the value
J_
1.2
/ (a) g (a) h (a)
where £ lies between a and 6, and 17 lies between a and c.
This proof does not presuppose the continuity of the second derivatives
f"(x), g"(x), h"(x). If these derivatives are continuous, and if the values a, 6, c
approach the same limit XQ, we have, in the limit,
1
/ (x0) g (x0) h (x0)
f (x0) g' (xo) h' (x0)
f"(x0) 0"(xo) h"(xQ)
Analogous expressions exist for n functions and the proof follows the same
lines. If only two functions /(x) and g (x) are taken, the formula? reduce to the
law of the mean if we set g (x) = 1.
An analogous generalization has been given by Schwarz (Annali di Mathe-
matica, 2d series, Vol. X).
II. FUNCTIONS OF SEVERAL VARIABLES
10. Introduction. A variable quantity w whose value depends on
the values of several other variables, x, y, z, ••-, t, which are in
dependent of each other, is called a function of the independ
ent variables x, y, z, •••, t; and this relation is denoted by writing
w =f(x, y,z,---, t). For definiteness, let us suppose that w = f(x, y)
is a function of the two independent variables x and y. If we think
of x and y as the Cartesian coordinates of a point in the plane,
each pair of values (x, y) determines a point of the plane, and con
versely. If to each point of a certain region A in the xy plane,
bounded by one or more contours of any form whatever, there
corresponds a value of w, the function f(x, y) is said to be defined
in the region A.
Let (x0, y0) be the coordinates of a point M0 lying in this region.
The function f(x, y) is said to be continuous for the pair of values
(xoi yo) if, corresponding to any preassigned positive number c, another
positive number 77 exists such that
|/C*o + h, y0 + k)-f(x0, 2/0) | < e
whenever \h < rj and \k\<rj.
This definition of continuity may be interpreted as follows. Let
us suppose constructed in the xy plane a square of side 2^ about
M0 as center, with its sides parallel to the axes. The point M',
12 DERIVATIVES AND DIFFERENTIALS [I, §11
whose coordinates are x0 + h, y0 + k, will lie inside this square, if
| h | < rj and | k \ < rj. To say that the function is continuous for the
pair of values (x0, T/O) amounts to saying that by taking this square
sufficiently small we can make the difference between the value of
the function at M0 and its value at any other point of the square less
than e in absolute value.
It is evident that we may replace the square by a circle about
(x0, y0) as center. For, if the above condition is satisfied for all
points inside a square, it will evidently be satisfied for all points
inside the inscribed circle. And, conversely, if the condition is
satisfied for all points inside a circle, it will also be satisfied for all
points inside the square inscribed in that circle. We might then
define continuity by saying that an rj exists for every c, such that
whenever V/i2 + k'2 < 17 we also have
I /(«<> + h> y0 + k) -f(x0,
The definition of continuity for a function of 3, 4, • • • , n inde
pendent variables is similar to the above.
It is clear that any continuous function of the two independent
variables x and y is a continuous function of each of the variables
taken separately. However, the converse does not always hold.*
11. Partial derivatives. If any constant value whatever be substi
tuted for y, for example, in a continuous function f(x, y), there
results a continuous function of the single variable x. The deriva
tive of this function of x, if it exists, is denoted by fx(x, y) or by <ax.
Likewise the symbol uv, or fy (x, y), is used to denote the derivative
of the function f(x, y} when x is regarded as constant and y as the
independent variable. The functions fx(x, y) and fy (x, y) are called
the partial derivatives of the function f(x, y). They are themselves,
in general, functions of the two variables x and y. If we form their
partial derivatives in turn, we get the partial derivatives of the sec
ond order of the given function f(x, y). Thus there are four partial
derivatives of the second order, fa (x, y),fx¥(x, y),fyx(x, y),f+(x, y\
The partial derivatives of the third, fourth, and higher orders are
* Consider, for instance, the f unction /(x, y), which is equal to 2 xy / (x2 + y2) when
the two variables x and y are not both zero, and which is zero when x = y = 0. It is
evident that this is a continuous function of x when y is constant, and vice versa.
Nevertheless it is not a continuous function of the two independent variables x and y
for the pair of values x = 0, y = 0. For, if the point (a-, y) approaches the origin upon
the line x = y. the f unction/ (x, y) approaches the limit 1, and not zero. Such functions
have been studied by Baire in his thesis.
I, §11] FUNCTIONS OF SEVERAL VARIABLES 13
defined similarly. In general, given a function w = /(x, y, z, ••-, f)
of any number of independent variables, a partial derivative of the
nth order is the result of n successive differentiations of the function
/, in a certain order, with respect to any of the variables which occur
in /. We will now show that the result does not depend upon the
order in which the differentiations are carried out.
Let us first prove the following lemma :
Let w = f (x, y) be a function of the two variables x and y. Then
fxij = ftjx, provided that these two derivatives are continuous.
To prove this let us first write the expression
U =f(x + Ax, y + Ay) -f(x, y + Ay) -f(x + Ax, y) + /(x, y}
in two different forms, where we suppose that x, y, Ax, A?/ have
definite values. Let us introduce the auxiliary function
<£ 00 =f(x + Ax, u) -/(x, v),
where v is an auxiliary variable. Then we may write
Applying the law of the mean to the function <£(w), we
U = Ay <£„ (y + 0Ay), where 0 < 0 < 1 ;
or, replacing <j>u by its value,
U = Ay [/„(* + Ax, y + 0Ay) -fy(x, y + 0Ay)].
If we now apply the law of the mean to the function fy (u, y + 0Ay),
regarding u as the independent variable, we find
U = Ax Ay/^ (x + 0' Ax, y + 0Ay), 0 < 0' < 1.
From the symmetry of the expression U in x, y, Ax, Ay, we see that
we would also have, interchanging x and y,
U = Ay Aaj/q, (x + 0| Ax, y + ^ Ay),
where 0, and 0[ are again positive constants less than unity. Equat
ing these two values of U and dividing by Ax Ay, we have
fxy(x + 0[ Ax, y + ^Ay) =f,,x(x + 0'Ax, y + 0Ay).
Since the derivatives /,.„ (x, y) and fvx(x, y) are supposed continuous,
the two members of the above equation approach fxy (x, y) and
fyx(x, y), respectively, as Ax and Ay approach zero, and we obtain
the theorem which we wished to prove.
14 DERIVATIVES AND DIFFERENTIALS [I, § n
It is to be noticed in the above demonstration that no hypothesis
whatever is made concerning the other derivatives of the second order,
f^ and fyt. The proof applies also to the case where the function
f(x, y) depends upon any number of other independent variables
besides x and y, since these other variables would merely have to
be regarded as constants in the preceding developments.
Let us now consider a function of any number of independent
variables,
«=/(»> y> *>•••> *)j
and let n be a partial derivative of order n of this function. Any
permutation in the order of the differentiations which leads to fi
can be effected by a series of interchanges between two successive
differentiations ; and, since these interchanges do not alter the
result, as we have just seen, the same will be true of the permuta
tion considered. It follows that in order to have a notation which
is not ambiguous for the partial derivatives of the nth order, it is
sufficient to indicate the number of differentiations performed with
respect to each of the independent variables. For instance, any nth
derivative of a function of three variables, to =/(x, y, z), will be
represented by one or the other of the notations
where p -f- q + r = n* Either of these notations represents the
result of differentiating / successively p times with respect to x,
q times with respect to ?/, and r times with respect to 2, these oper
ations being carried out in any order whatever. There are three
distinct derivatives of the first order, fx, f , fz\ six of the second
order, fa, fa fa /3.v, fa fxz ; and so on.
In general, a function of p independent variables has just as many
distinct derivatives of order n as there are distinct terms in a homo
geneous polynomial of order n in p independent variables ; that is,
as is shown in the theory of combinations.
Practical rules. A certain number of practical rules for the cal
culation of derivatives are usually derived in elementary books on
* The notation /a Pyq..r (x, y, z) is used instead of the notation fxfynzr (x, y, z) for
simplicity. Thus the notation fxy (x, y), used in place of f'x'y(x, y), is simpler and
equally clear. — TRANS.
I, §11] FUNCTIONS OF SEVERAL VARIABLES 16
the Calculus. A table of such rules is appended, the function and
its derivative being placed on the same line :
y' = ax0-'1;
y' = ax log a,
where the symbol log denotes the natural logarithm ;
y = log x, y' = ->
X
y = sin x,
y = cos x,
y •= arc sin x,
y = arc tan x,
JL -\- X
y = uv, y' = u'v 4- uv1 ;
_ u f ^ u'v — uv' .
y =/(«), 2/*=/>K;
The last two rules enable us to find the derivative of a function
of a function and that of a composite function if fu,fv,fw are con
tinuous. Hence we can find the successive derivatives of the func
tions studied in elementary mathematics, — polynomials, rational
and irrational functions, exponential and logarithmic functions,
trigonometric functions and their inverses, and the functions deriv
able from all of these by combination.
For functions of several variables there exist certain formulae
analogous to the law of the mean. Let us consider, for definite-
ness, a function f(x, y) of the two independent variables x and y.
The difference f(x + h, y 4- K) — f(x, y) may be written in the form
f(x + h,y + k) -f(x, y) = [/(* + h, y + k) -f(x, y + &)]
to each part of which we may apply the law of the mean. We
thus find
f(x + h,y + k}-f(x, y) = hfx(x + 6h, y + k}+ kfv(x, y + O'K),
where 6 and 0' each lie between zero and unity.
This formula holds whether the derivatives fx and /„ are continu
ous or not. If these derivatives are continuous, another formula,
16 DERIVATIVES AND DIFFERENTIALS [1,512
similar to the above, but involving only one undetermined number
6, may be employed.* In order to derive this second formula, con
sider the auxiliary function <f>(£) = f(x + ht, y + kfy, where x, y, h,
and k have determinate values and t denotes an auxiliary variable.
Applying the law of the mean to this function, we find
Now <£(>") is a composite function of t, and its derivative 4>'(t) is
equal to hfx (x -f- ht, y + kf) + kfy (x + ht, y -f- kt) ; hence the pre
ceding formula may be written in the form
12. Tangent plane to a surface. We have seen that the derivative
of a function of a single variable gives the tangent to a plane curve.
Similarly, the partial derivatives of a function of two variables occur
in the determination of the tangent plane to a surface. Let
(2) z . F(x, y)
be the equation of a surface S, and suppose that the function F(x, ?/),
together with its first partial derivatives, is continuous at a point
(^o? yo) of the xy plane. Let z0 be the corresponding value of z,
and AT0 (cr0, 7/0 > «0) the corresponding point on the surface S. If
the equations
(3) *=/(*), z/ = <KO> * = ^(9
represent a curve C on the surface S through the point M0, the
three functions f(f), <j>(t), "A(0> which we shall suppose continuous
and differentiable, must reduce to x0, y0, z0, respectively, for some
value t0 of the parameter t. The tangent to this curve at the point
M0 is given by the equations (§ 5)
x — x° Y — z — *
Since the curve C lies on the surface S, the equation \j/(t)=F[f(t~), .
must hold for all values of t; that is, this relation must be an identity
* Another formula may be obtained which involves only one undetermined number 0,
and which holds even when the derivatives/^, and/, are discontinuous. For the applica
tion of the law of the mean to the auxiliary function <j>(t) =f(x + ht,y + k) +f(x, y + kt)
gives
<£(!) -0(0) = 0'(0), 0<0<1.
or
f(x + h,y + k) -f(x, y) = hfx(x + 6h, y + k) + kfy(x, y + 6k), 0<0<1.
The operations performed, and hence the final formula, all hold provided the deriva
tives fx and fy merely exist at the points (x + ht, y + k), (x,y + kt),0^t^\. — TRANS.
I, § 13] FUNCTIONS OF SEVERAL VARIABLES 17
in t. Taking the derivative of the second member by the rule for
the derivative of a composite function, and setting t = t0, we have
(5) <j,'(t0)=fi(t0)FXo + <t>'(t0)FVa.
We can now eliminate f'(t0~), <£'(£0)> i//(£0) between the equations (4)
and (5), and the result of this elimination is
(6) Z-z0 = (X- ar0) FXg + (Y - y0) F^.
This is the equation of a plane which is the locus of the tangents to
all curves on the surface through the point M0. It is called the tan
gent plane to the surface.
13. Passage from increments to derivatives. We have defined the successive
derivatives in terms of each other, the derivatives of order n being derived from
those of order (n — 1), and so forth. It is natural to inquire whether we may
not define a derivative of any order as the limit of a certain ratio directly, with
out the intervention of derivatives of lower order. We have already done some
thing of this kind for fxy (§ 11); for the demonstration given above shows that/rj,
is the limit of the ratio
f(x + Ax, y + Ay) -/(x + Ax, y)-f(x, y + Ay) + /(x, y)
Ax Ay
as Ax and Ay both approach zero. It can be shown in like manner that the
second derivative /" of a function f(x) of a single variable is the limit of the
ratio
/(x + hi + h*) -f(x + hi) -f(x
^1^2
as hi and h2 both approach zero.
For, let us set
/i(x)=/(z + Ai)
and then write the above ratio in the form
h\
f>(x +
+
hi
The limit of this ratio is therefore the second derivative /", provided that
derivative is continuous.
Passing now to the general case, let us consider, for definiteness, a function of
three independent variables, w =f(x, y, 2). Let us set
A£W =/(x + h, y, z) -/(x, y, 2),
A£W =/(x, y + k, 2) -/(x, y, 2),
A^w =/(x, y, 2 -f 1) -/(x, y, z),
where A* w, A* w, Alz u are the^irsi increments of w. If we consider ^, k, I as given
constants, then these three first increments are themselves functions of x, y, 2,
and we may form the relative increments of these functions corresponding to
18 DERIVATIVES AND DIFFERENTIALS [I, § 13
increments hi, ki, ^ of the variables. This gives us the second increments,
A*1 A*w> A*1 Avw' ' ' ' • Tnis process can be continued indefinitely ; an increment
of order n would be defined as a first increment of an increment of order (n — 1).
Since we may invert the order of any two of these operations, it will be suffi
cient to indicate the successive increments given to each of the variables. An
increment of order n would be indicated by some such notation as the following :
A<->« = AX' ' ' ' A*p A*' • • • AX1 • • • A^/(z, y, z),
where p + q + r = n, and where the increments h, k, I may be either equal or
unequal. This increment may be expressed in terms of a partial derivative of
order n, being equal to the product
hihy • • • hpki • • • kgl\ • • • lr
x fxp*z'(x + *i Ai + • • • + d,,hp, y + eiki + • • • + O'qkq, z + ffi'li + • • • + Kir),
where every 6 lies between 0 and 1. This formula has already been proved for
first and for second increments. In order to prove it in general, let us assume
that it holds for an increment of order (n — 1), and let
0 (X, y, 2) = A** • • • Ah/ Ajt • • • AX1 ' ' ' £ f.
Then, by hypothesis,
$(x,y,z) = hz---hpki-- -kqli-- •Irfxp-iif,ir(x + 0sh2+ ---- \-6Php, y-\ ---- ,«H ---- ).
But the nth increment considered is equal to 0(x + hi, y, z) — <f>(x, y, z); and if we
apply the law of the mean to this increment, we finally obtain the formula sought.
Conversely, the partial derivative fxT^zr is the limit of the ratio
AX'- .-AX-.. -AX'--- A/-/
hi h? • • • hp ki k2 • • • kg li • • • lr
as all the increments h, k, I approach zero.
It is interesting to notice that this definition is sometimes more general than
the usual definition. Suppose, for example, that w =/(x, y) — <f>(x) + ^(y) is a
function of x and y, where neither <f> nor ^ has a derivative. Then u also has
no first derivative, and consequently second derivatives are out of the question,
in the ordinary sense. Nevertheless, if we adopt the new definition, the deriva
tive fxy is the limit of the fraction
/(x + h, y + k) -/(x + h, y) -/(x, y + k) +/(x, y)
hk
which is equal to
h) + t(V + k) - <t>(x + h) -
hk
But the numerator of this ratio is identically zero. Hence the ratio approaches
zero as a limit, and we find/xy = 0.*
* A similar remark may be made regarding functions of a single variable. For
example, the f unction /(K) = xs cosl/x has the derivative
f'(x) = 3 x2 cos - + xsin-i
and f'(x) has no derivative for x — 0. But the ratio
/(2ar)-2/(tt)+/(0)
o"
or 8 a cos (I/ 2 a) — 2 a cos (I/ or), has the limit zero when a approaches zero.
l)§14] THE DIFFERENTIAL NOTATION 19
III. THE DIFFERENTIAL NOTATION
The differential notation, which has been in use longer than any
other,* is due to Leibniz. Although it is by no means indispensable,
it possesses certain advantages of symmetry and of generality which
are convenient, especially in the study of functions of several varia
bles. This notation is founded upon the use of infinitesimals.
14. Differentials. Any variable quantity which approaches zero as
a limit is called an infinitely small quantity, or simply an infinitesi
mal. The condition that the quantity be variable is essential, for
a constant, however small, is not an infinitesimal unless it is zero.
Ordinarily several quantities are considered which approach zero
simultaneously. One of them is chosen as the standard of compari
son, and is called the principal infinitesimal. Let « be the principal
infinitesimal, and ft another infinitesimal. Then 0 is said to be an
infinitesimal of higher order with respect to a, if the ratio ft/a
approaches zero with a. On the other hand, ft is called an infini
tesimal of the first order with respect to a, if the ratio ft/a
approaches a limit K different from zero as a approaches zero. In
this case
^ = K + e,
«
where c is another infinitesimal with respect to a. Hence
ft=a(K + c)= Ka + at,
and Ka is called the principal part of ft. The complementary term
at is an infinitesimal of higher order with respect to a. In general,
if we can find a positive power of a, say a", such that ft /a"
approaches a finite limit K different from zero as a approaches
zero, ft is called an infinitesimal of order n with respect to a. Then
we have
4 ; = K + e,
a
or
ft = an (K -f e) = Ka* + «"«.
The term Ka" is again called the principal part of ft.
Having given these definitions, let us consider a continuous func
tion y=f(x), which possesses a derivative f'(x). Let Aa; be an
* With the possible exception of Newton's notation. — TRANS.
20 DERIVATIVES AND DIFFERENTIALS [I, § 14
increment of x, and let A?/ denote the corresponding increment of y.
From the very definition of a derivative, we have
where c approaches zero with Ace. If Ax be taken as the principal
infinitesimal, AT/ is itself an infinitesimal whose principal part is
f'(x) Ax.* This principal part is called the differential of y and is
denoted by dy.
dy=f(x)&x.
When /(x) reduces to x itself, the above formula becomes dx = Ax ;
and hence we shall write, for symmetry,
where the increment dx of the independent variable x is to be given
the same fixed value, which is otherwise arbitrary and of course
variable, for all of the several dependent
functions of x which may be under consid
eration at the same time.
Let us take a curve C whose equation is
y = f(x), and consider two points on it, M
and M', whose abscissae are x and x -f dx,
respectively. In the triangle MTN we have
NT = MN tan Z TMN = dxf'(x).
Hence NT represents the differential dy,
while Ay is equal to NM'. It is evident from the figure that M'T
is an infinitesimal of higher order, in general, with respect to NT,
as M' approaches M, unless MT is parallel to the x axis.
Successive differentials may be defined, as were successive deriv
atives, each in terms of the preceding. Thus we call the differ
ential of the differential of the first order the differential of the
second order, where dx is given the same value in both cases, as
above. It is denoted by d2y:
d*y = d (dy) = [/"(x) dx] dx = f"(x) (dx}*.
Similarly, the third differential is
d*y = d(d*y) = [_f»(x)dx*]dx =f"(x)(dx)*,
* Strictly speaking, we should here exclude the case where f'(x) = 0. It is, how
ever, convenient to retain the same definition of dy =f'(x)&x in this case also.
even though it is not the principal part of Ay. — TRANS.
I, §14] THE DIFFERENTIAL NOTATION 21
and so on. In general, the differential of the differential of order
(n — 1) is
The derivatives /'(or), /"(a), ••-, f(n\x), ... can be expressed, on
the other hand, in terms of differentials, and we have a new nota
tion for the derivatives :
t — dy ,, _ <Py M dny
y ~ dx' ~ dx2' ~dtft>
To each of the rules for the calculation of a derivative corresponds
a rule for the calculation of a differential. For example, we have
d xm = mxm-ldx, dax = ax log a dx ;
, , dx
d log x = — j d sin x = cos x dx ; • • • ;
SC
, . dx dx
aarcsmcc = •> darctanx = - -•
± Vl - a;2 1 + x2
Let us consider for a moment the case of a function of a function.
y =/(«), where u is a function of the independent variable x.
whence, multiplying both sides by dx, we get
yxdx =/(M) X uxdx;
that is,
dy =f(u)du.
The formula for dy is therefore the same as if u were the inde
pendent variable. This is one of the advantages of the differential
notation. In the derivative notation there are two distinct formulae,
&=/(*)> yx=f(u)uxy
to represent the derivative of y with respect to cc, according as y is
given directly as a function of x or is given as a function of x by
means of an auxiliary function u. In the differential notation the
same formula applies in each case.*
If y = f(u, v, w) is a composite function, we have
Vx = Uxfu + Vxfv + Wxfn,
at least if fu,fv,fw are continuous, or, multiplying by dx,
yxdx = uxdxfu + vxdxfv + wxdxfw ;
* This particular advantage is slight, however ; for the last formula ahove is equally
well a general one and covers both the cases mentioned. — TRANS.
22 DERIVATIVES AND DIFFERENTIALS [I, § 15
that is,
dll = fudu + fvdv +fwdw.
Thus we have, for example,
V du,
V
The same rules enable us to calculate the successive differentials.
Let us seek to calculate the successive differentials of a function
y = /(u), for instance. We have already
dy=f'(u}du.
In order to calculate d?y, it must be noted that du cannot be regarded
as fixed, since u is not the independent variable. We must then
calculate the differential of the composite function f'(u) du, where u
and du are the auxiliary functions. We thus find
To calculate d*y, we must consider d*y as a composite function, with
u, du, d2u as auxiliary functions, which leads to the expression
d*y =f'"(u)du8 + 3f"(u)dud*u +f'(u)d*u ;
and so on. It should be noticed that these formulae for d*y, d*y,
etc., are not the same as if u were the independent variable, on
account of the terms d*u, dzu, etc.*
A similar notation is used for the partial derivatives of a function
of several variables. Thus the partial derivative of order n of
f(x, y, s), which is represented by fxf>flzr in our previous notation,
is represented by
in the differential notation.f This notation is purely symbolic, and
in no sense represents a quotient, as it does in the case of functions
of a single variable. _ _ . __
15. Total differentials. Let w =f(x, y, z) be a function of the
three independent variables x, y, z. The expression
o /• o /» Q /«
du = ^- dx + ^ dy + -^ dz
ex dy - dz
* This disadvantage would seem completely to offset the advantage mentioned
above. Strictly speaking, we should distinguish between d^y and d?uy, etc. — TRANS.
t This use of the letter d to denote the partial derivatives of a function of several
variables is due to Jacob! . Before his time the same letter d was used as is used for
the derivatives of a function of a single variable.
I, §15] THE DIFFERENTIAL NOTATION 23
is called the total differential of o>, where dx, dy, dz are three fixed
increments, which are otherwise arbitrary, assigned to the three
independent variables x, y, z. The three products
8f 7 df ' j df j
TT- dx. £- dy, ~ dz
ex dy ' cz
are called partial differentials.
The total differential of the second order d*<a is the total differ
ential of the" total differential of the first order, the increments
dx, dy, dz remaining the same as we pass from one differential to
the next higher. Hence
_ , 7 . ddia ddia cdw
d2u = d(dta) = — — dx -f -^ — dy + -=— dz ;
Ox oy cz
or, expanding,
ex* dx oy ex cz
!
Oy Oz
+ 2 - dxdy + 2 dxdz + 2 •=•• dy dz.
Ox oy ox cz Oy Oz
If cPf be replaced by df2, the right-hand side of this equation
becomes the square of
We may then write, symbolically,
0x cy oz
it being agreed that df* is to be replaced by 82f after expansion.
In general, if we call the total differential of the total differential
of order (n —1) the total differential of order n, and denote it by
dn(a, we may write, in the same symbolism,
*.-(£**»* +£*)",
\0x Oy Oz /
where dfn is to be replaced by dnf after expansion ; that is, in our
ordinary notation,
DERIVATIVES AND DIFFERENTIALS [I, §15
where
A n'
pqr p\q\r\
is the coefficient of the term ap&'cr in the development of (a. + b + c)n.
For, suppose this formula holds for dn w. We will show that it then
holds for dn+l<o; and this will prove it in general, since we have
already proved it for n = 2. From the definition, we find
dn+l w=d(dn(o)
r zn+if dn+lf
- +
whence, replacing en + 1/by cfn + l, the right-hand side becomes
f ( 7T- dx -f 7f- dy + — rf« I ,
1 C7« ^V <7«
or
cy cz I \ox cy
Hence, using the same symbolism, we may write
- -
cy ' cz
Note. Let us suppose that the expression for dw, obtained in any
way whatever, is
(7) dw = P dx -f- Q dy + R dz,
where P, Q, R are any functions x, y, z. Since by definition
d<a 8<a d<a
rfw = ^- <£c + — ay + ^- dz,
dx cy cz
we must have
where dx, dy, dz are any constants. Hence
/«\ S<a - P go) - n 8<a - P
(o) "5~ — - .r, ^~ — y, "5~~ — -ft.
^X ^ KB
The single equation (7) is therefore equivalent to the three separate
equations (8) ; and it determines all three partial derivatives at once.
I, §16] THE DIFFERENTIAL NOTATION 25
In general, if the nth total differential be obtained in any way
whatever,
d" w = 2 Cpqr dx" dy" dzr ;
then the coefficients Cyqr are respectively equal to the corresponding
nth derivatives multiplied by certain numerical factors. Thus all
these derivatives are determined at once. We shall have occasion
to use these facts presently.
16. Successive differentials of composite functions. Let w = F(u, v, w~)
be a composite function, u, v, w being themselves functions of the
independent variables x, y, z, t. The partial derivatives may then be
written down as follows :
dia_dFdii dFdv dFdw
dx du dx dv dx dw dx
d«> _d_F_d_u d_F_d_v_ ___
dy du dy dv dy dw dy
d_F_d_v
dz du dz do dz dw dz
dw _ dF du dF dv dF dw
dt du dt dv dt dw dt
If these four equations be multiplied by dx, dy, dz, dt, respectively,
and added, the left-hand side becomes
d(» , <?W 7 , ^<° 7 ,^WJ.
•3- dx + -r- d y + -^- dz + -^ dt,
dx dy dz d£
that is, do* ; and the coefficients of
d_F d]F 0F
du do dw
on the right-hand side are du, dv, dw, respectively. Hence
dF dF dF
(9) do) = ^— du + -r— dv + ^— dw,
cu dv cw
and we see that the expression of the total differential of the first
order of a composite function is the same as if the auxiliary functions
were the independent variables. This is one of the main advantages
of the differential notation. The equation (9) does not depend, in
form, either upon the number or upon the choice of the independent
variables ; and it is equivalent to as many separate equations as
there are independent variables.
To calculate d2w, let us apply the rule just found for dta, noting
that the second member of (9) involves the six auxiliary functions
u, v, w, du, dv, dw. We thus find
26 DERIVATIVES AND DIFFERENTIALS [I, § lu
d2F dF
= -i^—- du2 4- -z — — du dv + - — — du dw + -^- dzu
Ctr cucu cucw en
4- - dudv 4- ^ dv2 + ff dvdw + ^
du dv cv2 cu cw cv
d2F d2F d'2F dF
+ „ „ du dw 4- o Q dv dw -f TT-^ t?w + ^—
^«gw ^y^M> Cw1 cw
or, simplifying and using the same symbolism as above,
7 , ^ , ^ ^ , „
d2w = [7^- du+ ^- dv + — dw\ + TT- »*« + — c?2w 4- ^— «•«•.
Vc/w ^y CM; / cu Co cw
This formula is somewhat complicated on account of the terms in
d2u, dzv, dzw, which drop out when u, v, w are the independent
variables. This limitation of the differential notation should be
borne in mind, and the distinction between d2w in the two cases
carefully noted. To determine ds w, we would apply the same rule
to <22o>, noting that d2w depends upon the nine auxiliary functions
u, v, w, du, dv, dw, d2 u, d2 v,d2w; and so forth. The general expres
sions for these differentials become more and more complicated ;
dnw is an integral function of du, dv, dw, d'2u, • • •, dnu, dnv, dnw, and
the terms containing dnu, dnv, dnw are
dF 7 dF , dF 7
— dnu 4- — dn v 4- — dnw.
cu cv cw
If, in the expression for d" w, u, v, w, du, dv, dw, • • • be replaced by
their values in terms of the independent variables, dnt» becomes an
integral polynomial in dx, dy, dz, • • • whose coefficients are equal
(cf. Note, § 15) to the partial derivatives of w of order n, multiplied
by certain numerical factors. We thus obtain all these derivatives
at once.
Suppose, for example, that we wished to calculate the first and
second derivatives of a composite function <a=f(ii), where w is a
function of two independent variables u = <f> (x, y). If we calculate
these derivatives separately, we find for the two partial derivatives
of the first order
1ft 8 w _ 8 w d u 8u) _ du> du
dx du dx dy du dy
Again, taking the derivatives of these two equations with respect
to x, and then with respect to y, we find only the three following
distinct equations, which give the second derivatives :
THE DIFFERENTIAL NOTATION
27
(11)
dx*
dx dy
du\*
ex
du
d*
0 <i> 0 U C U C <a
du* Cx dy du dx dy
d- 22
da dy
&ut
,2*
The second of these equations is obtained by differentiating the
first of equations (10) with respect to y, or the second of them with
respect to x. In the differential notation these five relations (10)
and (11) may be written in the form
(12)
en
cu
If du and d*u in these formulae be replaced by
'du
TT- dy and -^—a
dx dy
respectively, the coefficients of dx and dy in the first give the first
partial derivatives of «o, while the coefficients of dxz, 2 dx dy, and
dy2 in the second give the second partial derivatives of w.
17. Differentials of a product. The formula for the total differential
of order n of a composite function becomes considerably simpler
in certain special cases which often arise in practical applications.
Thus, let us seek the differential of order n of the product of two
functions o> = uv. For the first values of n we have
dw = v dti + u dv, d* a) = v d* u + 2 du dv -f ud* v, • • • ;
and, in general, it is evident from the law of formation that
d" w = v d" u 4- r, dr dn~^u + C»d*v dn~2n -f • • • +
where Clt C2, • • • are positive integers. It might be shown by alge
braic induction that these coefficients are equal to those of the
expansion of (a + &)" ; but the same end may be reached by the
following method, which is much more elegant, and which applies
to many similar problems. Observing that Cl, C2, • • • do not depend
upon the particular functions n and v employed, let us take the
28 DERIVATIVES AND DIFFERENTIALS [I, § 17
special functions u = e*, v = &, where x and y are the two inde
pendent variables, and determine the coefficients for this case. We
thus find
w = ex+y, dw = ex+y(dx + dy), • • -, dn<* = ex + y(dx + dy)n,
du = e*dx, dzu = exdx*, •• •,
dv — evdy, d*v = eydy2, • • •;
and the general formula, after division by ex+'J, becomes
(dx + di/}n = dx* + C^dydx*-1 + Ctdy2dxn-2 -\ [-dp.
Since dx and dy are arbitrary, it follows that
r _n n(n-l) n(n -1) -• • (n - p + 1)
Cl~l' 1.2 "' p~ 1.2-..p
and consequently the general formula may be written
(13) dn(uv) = vdnu+^dudn-lu + 7^ ^ d2vdn~2u -\ \-ud*v.
1 1 . 4
This formula applies for any number of independent variables.
In particular, if u and v are functions of a single variable x, we
have, after division by dxn, the expression for the nth derivative of
the product of two functions of a single variable.
It is easy to prove in a similar manner formulae analogous to
(13) for a product of any number of functions.
Another special case in which the general formula reduces to a
simpler form is that in which u, v, w are integral linear functions
of the independent variables x, y, z.
u= ax -f by + cz+f,
v = a'x + b'y + c'z +/' ,
w = a"x + b"y + c"z +/",
where the coefficients a, a', a", b, b', • • • are constants. For then we
have
du = a dx + b dy + c dz,
dv = a' dx -f- b'dy + c'dz,
dw = a"dx + b"dy + c"dz,
and all the differentials of higher order dnu, dnv, dniv, where n>l,
vanish. Hence the formula for dn<j> is the same as if u, v, w were
the independent variables ; that is,
I, §18] THE DIFFERENTIAL NOTATION 29
(dF . 8F . 8F , V">
dnw = -5- du + -T- dv 4- 5— dw I .
We proceed to apply this remark.
18. Homogeneous functions. A function <f>(x, y, z) is said to be
homogeneous of degree m, if the equation
<£(w, v, w)= tm$(x, y, z)
is identically satisfied when we set
u = tx, v = ty, w = tz.
Let xis equate the differentials of order n of the two sides of this
equation with respect to t, noting that u, v, w are linear in t, and that
du = x dt, dv = y dt, dw = z dt.
The remark just made shovvs that
ihi + y dfo + *^)(">== m(m ~1} ' " (m " n '+1)<m""*(a;' y»*)-
If we now set # = 1, w, v, w reduce to #, ?/, 2, and any term of
the development of the first member,
becomes
d"<>
whence we may write, symbolically,
which reduces, for n = 1, to the well-known formula
Various notations. We have then, altogether, three systems of nota
tion for the partial derivatives of a function of several variables, —
that of Leibniz, that of Lagrange, and that of Cauchy. Each of
these is somewhat inconveniently long, especially in a complicated
calculation. For this reason various shorter notations have been
devised. Among these one first used by Monge for the first and
30
DERIVATIVES AND DIFFERENTIALS
[I, §19
second derivatives of a function of two variables is now in common
use. If z be the function of the two variables x and y, we set
P —
t =
dy ex2 ex 8y o if
and the total differentials dz and d'2z are given by the formulae
dz = p dx + q dy,
d2z = r dx'2 -f- 2 s dx dy + t dy~.
Another notation which is now coming into general use is the
following. Let z be a function of any number of independent vari
ables x1} xz, x3) • ••, xn ; then the notation
ex l ex.2 • • • ox
is used, where some of the indices alt a.2) • • •, an may be zeros.
19. Applications. Let y — f(x) be the equation of a plane curve C with
respect to a set of rectangular axes. The equation of the tangent at a point
M(x, y) is
Y-y = y'(X-x).
The slope of the normal, which is perpendicular to the tangent at the point of
tangency, is — l/y'; and the equation of the normal is, therefore,
Let P be the foot of the ordinate of the point Jlf, and let T and N be the
points of intersection of the x axis with the tangent and the normal, respectively.
The distance PN is called the subnormal ;
FT, the subtangent; MN, the normal; and
M T, the tangent.
From the equation of the normal the ab
scissa of the point N is x + yy', whence the
subnormal is ± yy'. If we agree to call the
length PN the subnormal, and to attach the
sign + or the sign — according as the direc
tion PN is positive or negative, the subnormal
will always be yy' for any position of the curve
C. Likewise the subtangent is — y /y'.
The lengths MN and M T are given by the triangles MPN and MPT:
Various problems may be given regarding these lines. Let us find, for
instance, all the curves for which the subnormal is constant and equal to a given
number a. This amounts to finding all the functions y=f(x) which satisfy
the equation yy' = a. The left-hand side is the derivative of 2/2/2, while the
I, EXS.] EXERCISES 31
right-hand side is the derivative of ax. These functions can therefore differ
only by a constant ; whence
y'2 = 2ax + C,
which is the equation of a parabola along the x axis. Again, if we seek the
curves for which the subtangent is constant, we are led to write down the equa
tion y'/y = l/«; whence
log2/ = - + logC, or y = Ce?,
a
which is the equation of a transcendental curve to which the x axis is an asymp
tote. To find the curves for which the normal is constant, we have the equation
or
/a2 - y*
The first member is the derivative of - Vo^- y2 ; hence
(x + C)2 + y2 = a2,
which is the equation of a circle of radius a, whose center lies on the x axis.
The curves for which the tangent is constant are transcendental curves, which
we shall study later.
Let y = f(x) and Y — F(x) be the equations of two curves C and C", and let
M, M' be the two points which correspond to the same value of x. In order that
the two subnormals should have equal lengths it is necessary and sufficient that
YY'=±yy';
that is, that Y2 — ± y2 + C, where the double sign admits of the normals' being
directed in like or in opposite senses. This relation is satisfied by the cirfves
and also by the curves
which gives an easy construction for the normal to the ellipse and to the hyperbola.
EXERCISES
1. Let p = f(6) be the equation of a plane curve in polar coordinates. Through
the pole O draw a line perpendicular to the radius
vector OM, and let T and N be the points where this
line cuts the tangent and the normal. Find expres
sions for the distances OT, ON, MN, and MT in
terms of /(0) and /'(<?).
Find the curves for which each of these distances,
in turn, is constant.
2. Let y = f(x), z—<t>(x) be the equations of a
skew curve T, i.e. of a general space curve. Let N FIG. 3
32 DERIVATIVES AND DIFFERENTIALS [I, Exs.
be the point where the normal plane at a point Af, that is, the plane perpendicu
lar to the tangent at .M", meets the z axis ; and let P be the foot of the perpen
dicular from M to the z axis. Find the curves for which each of the distances
PN and JOT, in turn, is constant.
[Note. These curves lie on paraboloids of revolution or on spheres.]
3. Determine an integral polynomial /(z) of the seventh degree in x, given
that f(x) + 1 is divisible by (x - I)4 and f(x) - 1 by (x+1)*. Generalize the
problem.
4. Show that if the two integral polynomials P and Q satisfy the relation
Vl -p-t = Q Vl - x2,
then
dP ndx
Vl - p* Vl - x2
where n is a positive integer.
[Note. From the relation
(a) l-P2 = Q2(l-x»)
it follows that
(b) - 2 PP' = Q [2 Q'(l - x*) - 2 Qx].
The equation (a) shows that Q is prime to P ; and (b) shows that P' is divisible
by Q-]
5*. Let E (x) be a polynomial of the fourth degree whose roots are all dif
ferent, and let x = U / V be a rational function of t, such that
where R\ (t) is a polynomial of the fourth degree and P / Q is a rational function.
Show that the function U/ V satisfies a relation of the form
dx kdt _ •
VR(X) Vfl!(«)'
where A; is a constant. [JACOBI.]
[Note. Each root of the equation R(U/ V) = 0, since it cannot cause R'(x)
to vanish, must cause UV — VU', and hence also dx/dt, to vanish.]
6*. Show that the nth derivative of a function y = $ (u), where u is a func
tion of the independent variable x, may be written in the form
where
1.2
~ (*=1, 2,
ctx
[First notice that the nth derivative may be written in the form (a), where the
coefficients AI, A*, • • -, An are independent of the form of the function <j>(u).
I, EXS.] EXERCISES 33
To find their values, set 0 (M) equal to w, «2, • • • , un successively, and solve the
resulting equations for Ait A^, • • • , An. The result is the form (b).]
7*. Show that the nth derivative of <f> (x2) is
"2 (»>(x2) + n(n -
dxn
+ n(n-V--(n-*P+V (2 X)n- 2P 0("-P)(x2) + • • • ,
1 . ju • • • p
where p varies from zero to the last positive integer not greater than n/2, and
where 0(0 (x2) denotes the ith derivative with respect to x.
Apply this result to the functions er3?, arc sin x, arc tan x.
8*. If x = cos u, show that
d»-i(l -x2)">-* , 1.3.5- • -(2m- 1) .
S ' = (— l)m~i — sin mu.
dxm~l m
[OLINDE RODRIGUES.]
9. Show that Legendre's polynomial,
2 . 4 . 6 • • • 2 n dx"
satisfies the differential equation
ax-1 ax
Hence deduce the coefficients of the polynomial.
10. Show that the four functions
yt = sin (n arc sin x), 2/3 = sin (n arc cos x),
y2 = cos (n arc sin x), 2/4 = cos (n arc cos x),
satisfy the differential equation
(1 - x2) y" - xy' + ri*y = Q.
Hence deduce the developments of these functions when they reduce to poly
nomials.
11*. Prove the formula
i
dn G*
_(x«-iei) = (-!)" -•
dx»V x»+!
[HALPHEN.]
12. Every function of the form z = x$(y/x) + $ (y/x) satisfies the equation
rx2 + 2 sxy + ty* = 0,
whatever be the functions <f> and ^.
13. The function z = x0(x + y) + y^(x + y) satisfies the equation
r - 2 s + t = 0,
whatever be the functions 0 and \f/.
34 DERIVATIVES AND DIFFERENTIALS [I, Exs
14. The function z =f[x + </>(y)] satisfies the equation ps = qr, whatever
be the functions / and 0.
15. The function z = x»<j>(y/x) + y~n^(y/x) satisfies the equation
rx2 + 2 sxy + ty2 + px + qy = n2z,
whatever be the functions <j> and \f/.
16. Show that the function
y - x - ai | 0! (x) + x - az \ fa (x) + • • • + | x - an \ 0n (x),
where fa (x), fa (x), • • • , 0n (x), together with their derivatives, 0i (x), 0£ (x), • • • ,
0n(x), are continuous functions of x, has a derivative which is discontinuous
for x = a\ , Oz , • • • , an •
17. Find a relation between the first and second derivatives of the function
« =/(&!» M), where M = 0(x2, x3); xt, x2, x3 being three independent variables,
and /and 0 two arbitrary functions.
18. Let/"(x) be the derivative of an arbitrary f unction /(x). Show that
1 d*u, _ 1 #2»
u dx2 v dx2'
where u = [/'(x)]-i and » =/(x) [/'(x)]-*.
19*. The nth derivative of a function of a function u-<p(y), where y = ^ (x),
may be written in the form
^1.2,
where the sign of summation extends over all the positive integral solutions of
the equation i + 2 j + 3 h -\ + Ik = n, and where p = i + j + • • . + k.
[FA A DE BRUNO, Quarterly Journal of Mathematics, Vol. I, p. 359.]
CHAPTER II
IMPLICIT FUNCTIONS FUNCTIONAL DETERMINANTS
CHANGE OF VARIABLE
I. IMPLICIT FUNCTIONS
20. A particular case. We frequently have to study functions for
which no explicit expressions are known, but which are given by
means of unsolved equations. Let us consider, for instance, an
equation between the three variables x, y, z,
(1) F(x, y, z) = 0.
This equation defines, under certain conditions which we are about
to investigate, a function of the two independent variables x and y.
We shall prove the following theorem :
Let x = x0, y =. y0, z = z0 b& a set of values which satisfy the equa
tion (1), and let us suppose that the function F, together with its first
derivatives, is continuous in the neighborhood of this set of values*
If the derivative Fz does not vanish for x = x0, y = y0, z = z0, there
exists one and only one continuous function of the independent variables
x and y which satisfies the equation (1), and which assumes the value z0
when x and y assume the values x0 and y0, respectively.
The derivative Fz not being zero for x = x0, y = y0, z = z0, let us
suppose, for defmiteness, that it is positive. Since F, Fx, Fv, Fz are
supposed continuous in the neighborhood, let us choose a positive
number I so small that these four functions are continuous for all
sets of values x, y, z which satisfy the relations
(2) \x-x0\<l, \y-y0\<l, \*-z0<l,
and that, for these sets of values of x, y, z,
Fz(x,y,z}>P,
*Iu a recent article (Bulletin de la Societe Mathematique de France, Vol. XXXI,
190.'?, pp. 184-192) Goursat has shown, by a method of successive approximations, that
it is not necessary to make any assumption whatever regarding Fx and Ft/, even as to
their existence. His proof makes no use of the existence of Fx and Fy. His general
theorem and a sketch of his proof are given in a footnote to § 25. — TRANS.
35
36 FUNCTIONAL RELATIONS [II, §20
where P is some positive number. Let Q be another positive num
ber greater than the absolute values of the other two derivatives
Fx, Fy in the same region.
Giving x, y, z values which satisfy the relations (2), we may then
write down the following identity :
F(*> V, *) - F(*o, 7/0, «0) = F(x, y, z} - F(x0, y, z) + F(x0, y, z)
-F(x0, T/o, z) +F(x0, 7/0j z) — F(x0, 7/0, «0) 5
or, applying the law of the mean to each of these differences, and
observing that F(x0, y0, «0) = 0,
F(x>y>*) = (* — *o)-FTar[»o + 8(x — »o), y, *]
+ 0 - yo) Fv [*., 2/o + ff(y - y0), «]
+ (z - «0) F2 [>o, ?/o, *o + 0"0 - «0)].
Hence -F(cc, T/, 2) is of the form
(3) S F^' y' ^ = A (x> y> ^ (x ~ xd
I +B(x, y, z) (y - T/O) + C (x, y, z) (z - *0),
where the absolute values of the functions A(x, y, z), B(x, y, z),
C(x, y, z) satisfy the inequalities
M|<Q, \B\<Q, \C\>P
for all sets of values of x, y, z which satisfy (2). Now let c be a
positive number less than Z, and rj the smaller of the two numbers
I and Pe/2Q. Suppose that x and y in the equation (1) are given
definite values which satisfy the conditions
and that we seek the number of roots of that equation, z being
regarded as the unknown, which lie between z0 — e and z0 + c. In
the expression (3), for F(x, y, z} the sum of the first two terms is
always less than 2Qrj in absolute value, while the absolute value of
the third term is greater than Pe when z is replaced by z0 ± e. From
the manner in which 77 was chosen it is evident that this last term
determines the sign of F. It follows, therefore, that F(x, y, z0 — e) < 0
and F(x, y, z0 + e) > 0 ; hence the equation (1) has at least one root
which lies between z0 — e and z0 + e. Moreover this root is unique,
since the derivative Fz is positive for all values of z between z0 — e
and z0 + e. It is therefore clear that the equation (1) has one and
only one root, and that this root approaches ZQ as x and y approach
XQ and ?/0, respectively.
II, § 20]
IMPLICIT FUNCTIONS
37
Let us investigate for just what values of the variables x and y
the root whose existence we have just proved is denned. Let h be
the smaller of the two numbers I and PI/2Q; the foregoing reason
ing shows that if the values of the variables x and y satisfy the
inequalities \x — x^\
< h, the equation (1) will have one
and only one root which lies between z0 — I and z0 -f I- Let R be a
square of side 2 h, about the point M0(x0, y0), with its sides parallel
to the axes. As long as the point (x, y) lies inside this square,
the equation (1) uniquely determines a function of x and y, which
remains between z0 — I and z0 + I. This function is continuous, by
the above, at the point M0, and this is likewise true for any other
point Ml of R; for, by the hypotheses made regarding the func
tion F and its derivatives, the derivative Ft(xlf yl} «i) will be posi
tive at the point Mlt since \xl — x0<l, \y\ — ya\<l, \zi~ zo\<l-
The condition of things at Ml is then exactly the same as at M0,
and hence the root under consideration will be continuous for
Since the root considered is defined only in the interior of the
region R, we have thus far only an element of an implicit function.
In order to define this function out
side of R, we proceed by successive
steps, as follows. Let L be a con
tinuous path starting at the point
(x0, y0~) and ending at a point (X, F)
outside of R. Let us suppose that
the variables x and y vary simul
taneously in such a way that the ~
point (x, y) describes the path L.
If we start at (x0, y0) with the value
z0 of z, we have a definite value of this root as long as we remain
inside the region R. Let M1(xl, y^ be a point of the path inside R,
and zt the corresponding value of z. The conditions of the theorem
being satisfied for x = xlt y = yl} z = zv, there exists another region
Rl} about the point MI, inside which the root which reduces to zl for
x = Xi, y = yi is uniquely determined. This new region #! will
have, in general, points outside of R. Taking then such a point Mt
on the path L, inside R± but outside R, we may repeat the same con
struction and determine a new region R2, inside of which the solu
tion of the equation (1) is defined; and this process could be
repeated indefinitely, as long as we did not find a set of values of
x, y, z for which Fz = 0. We shall content ourselves for the present
Fm 4
38 FUNCTIONAL RELATIONS [II, § 21
with these statements; we shall find occasion in later chapters to
treat certain analogous problems in detail.
21. Derivatives of implicit functions. Let us return to the region
R, and to the solution z = <f>(x, y) of the equation (1), which is a
continuous function of the two variables x and y in this region.
This function possesses derivatives of the first order. For, keeping
y fixed, let us give x an increment Ax. Then z will have an incre
ment Az, and we find, by the formula derived in § 20,
F(x + As, y,z + A«) - F(x, ij, z)
= Az Fx (x + 0Az, y,e + Az) -f Aa Ft (x, y, z + 0'Az) = 0.
Hence
and when A# approaches zero, As does also, since z is a continuous
function of a;. The right-hand side therefore approaches a limit,
and z has a derivative with respect to x :
In a similar manner we find
If the equation F = 0 is of degree m in z, it defines m
functions of the variables x and y, and the partial derivatives cz/cx,
3z/dy also have m values for each set of values of the variables
x and y. The preceding formulas give these derivatives without
ambiguity, if the variable z in the second member be replaced by
the value of that function whose derivative is sought.
For example, the equation
defines the two continuous functions
+ Vl — x'2 — y* and — Vl — x'2 — y2
for values of x and y which satisfy the inequality x- + y2 < 1.
The first partial derivatives of the first are
- y
II, §±2] IMPLICIT FUNCTIONS 39
and the partial derivatives of the second are found by merely chang
ing the signs. The same results would be obtained by using the
formulae
dz _ x Cz _ y
dx z cy z
replacing z by its two values, successively.
22. Applications to surfaces. If we interpret x, y, z as the Cartesian
coordinates of a point in space, any equation of the form
(4) F(x,y, z)=0
represents a surface S. Let (cc0, y0, z^) be the coordinates of a point
A of this surface. If the function F, together with its first deriva
tives, is continuous in the neighborhood of the set of values x0, yw z0,
and if all three of these derivatives do not vanish simultaneously
at the point A, the surface S has a tangent plane at A. Suppose,
for instance, that Fz is not zero for x = x0, y = y0, z = «0. Accord
ing to the general theorem we may think of the equation solved
for z near the point A, and we may write the equation of the surface
in the form
z = 4(x, y},
where <f> (x, y) is a continuous function ; and the equation of the
tangent plane at A is
Replacing dz /dx and dz /dy by the values found above, the equation
of the tangent plane becomes
If Fz = 0, but Fx =£ 0, at A , we would consider y and z as inde
pendent variables and a; as a function of them. We would then
find the same equation (5) for the tangent plane, which is also evi
dent a priori from the symmetry of the left-hand side. Likewise
the tangent to a plane curve F(x, y) = 0, at a point (x0, y0~), is
If the three first derivatives vanish simultaneously at the point A.
dF
40 FUNCTIONAL RELATIONS [II, §23
the preceding reasoning is no longer applicable. We shall see later
(Chapter III) that the tangents to the various curves which lie on
the surface and which pass through A form, in general, a cone and
not a plane.
In the demonstration of the general theorem on implicit functions
we assumed that the derivative F^ did not vanish. Our geometrical
intuition explains the necessity of this condition in general. For,
if F^ = 0 but F^ 3= 0, the tangent plane is parallel to the % axis,
and a line parallel to the z axis and near the line x = xw y = y0
meets the surface, in general, in two points near the point of
tangency. Hence, in general, the equation (4) would have two
roots which both approach z0 when x and y approach x0 and y0,
respectively.
If the sphere a;2 + y2 -+- «2 — 1 = 0, for instance, be cut by the line
y = 0, x = 1 + c, we find two values of z, which both approach zero
with e ; they are real if c is negative, and imaginary if c is positive.
23. Successive derivatives. In the formulae for the first derivatives,
3z= _Fx dz_ = _FJL
dx~ F,' cy~ F,'
we may consider the second members as composite functions, z being
an auxiliary function. We might then calculate the successive deriv
atives, one after another, by the rules for composite functions. The
existence of these partial derivatives depends, of course, upon the
existence of the successive partial derivatives of F(x, y, K).
The following proposition leads to a simpler method of determin
ing these derivatives.
If several functions of an independent variable satisfy a relation
F = 0, their derivatives satisfy the equation obtained by equating to
zero the derivative of the left-hand side formed by the rule for differ
entiating composite functions. For it is clear that if F vanishes
identically when the variables which occur are replaced by func
tions of the independent variable, then the derivative will also van
ish identically. The same theorem holds even when the functions
which satisfy the relation F = 0 depend upon several independent
variables.
Now suppose that we wished to calculate the successive derivatives
of an implicit function y of a single independent variable x defined
by the relation
II, §23] IMPLICIT FUNCTIONS 41
We find successively
dF
T~ +
ox cy
dF SF ,
~ ~
d'2F dF
—— + 2 — — v' + — — v'2+ —v" =0
* ^ dxdyy -dy» ^ dyy
2
3
dx* ox2 dy y ox dy* * dxdy dys
32 7,1
from which we could calculate successively y', y", y'
Example. Given a function y =/(x), we may, inversely, consider y as the
independent variable and x as an implicit function of y defined by the equation
y=f(x). If the derivative /'(x) does not vanish for the value XQ, where
2/o =/(zo)i there exists, by the general theorem proved above, one and only one
function of y which satisfies the relation y = f(x) and which takes on the value
XQ for y = 2/0- This function is called the inverse of the f unction /(x). To cal
culate the successive derivatives xy, xyt, ay, • • • of this function, we need merely
differentiate, regarding y as the independent variable, and we get
1 = /'(x) xy,
0 = /"(x) (X,)2 + /'(x) ay,
0 =/"'(x) (xy)* + 3f"(x)xyx? +/'(x)x2/3,
whence
1 f"(z) _8[/"(x)]«-
~7^)' ~[7w' [/'(
It should be noticed that these formulae are not altered if we exchange xv and
/'(x), Xy2 and /"(x), Xj,s and /"'(x), • • • , for it is evident that the relation between
the two functions y = /(x) and x = 0 (y) is a reciprocal one.
As an application of these formulae, let us determine all those functions
y=f(x) which satisfy the equation
y'y'" - 3y"* = 0.
Taking y as the independent variable and x as the function, this equation
becomes
Xj/> = 0.
But the only functions whose third derivatives are zero are polynomials of at
most the second deree. Hence x must be of the form
where Ci, C2, C3 are three arbitrary constants. Solving this equation for y,
we see that the only functions y = /(x) which satisfy the given equation are
of the form _
y = a ± V bx + c,
42 FUNCTIONAL RELATIONS [II, §24
where a, 6, c are three arbitrary constants. This equation represents a parabola
whose axis is parallel to the x axis.
24. Partial derivatives. Let us now consider an implicit function
of two variables, denned by the equation
(6) F(x,y,z) = 0.
The partial derivatives of the first order are given, as we have seen,
by the equations
9F.9F9* ?l dFdz_
(7) o — p 7T- — — u, -^ h fl ;p — u.
0x 9* 0z <?y </* 0#
To determine the partial derivatives of the second order we need
only differentiate the two equations (7) again with respect to x and
with respect to y. This gives, however, only three new equations,
for the derivative of the first of the equations (7) with respect to y
is identical with the derivative of the second with respect to x.
The new equations are the following:
.O±JL.?£ + ££/!?)%-—— =o
dx2 dxdzdx dz2 \dx] dz dx2
d2F , d*F dz ^ d2F d_z_ <P_F dz dz d_F d*z _
"" r\ n r\ o I O O O ^J
dx ftr dx dy c~
d2F d2F dz d2 F (dz\2 dF d2z
~ + " fe dy + 'dz2 (dy) +^ dy2 :
The third and higher derivatives may be found in a similar manner.
By the use of total differentials we can find all the partial deriva
tives of a given order at the same time. This depends upon the
following theorem :
If several functions u, v, w, • • • of any number of independent vari
ables x, y, z, ••• satisfy a relation F = 0, the total differentials satisfy
the relation dF= 0, which is obtained by forming the total differential
of F as if all the variables which occur in F ivere independent variables.
In order to prove this let F(u, v, w) = 0 be the given relation between
the three functions u, v, iv of the independent variables x, y, z, t. The
first partial derivatives of M, v, w satisfy the four equations
dFdu
__ __
du dx dv dx dw dx
d_Fd_u ,
du dy dv dy dw dy
II,§24J IMPLICIT FUNCTIONS 43
dFdu dFdv d_F_d-w_
o ~ « ~o ~ I o ~a — ^)
CM tfS 0» tf£ OW CZ
dFdu d_Fd_» ^^! = 0
aw & a? a< a^ st =
Multiplying these equations by dx, dy, dz, dt, respectively, and
adding, we find
°-^du + ^-dv + d-/-dw = dF=0.
du dv OW
This shows again the advantage of the differential notation, for the
preceding equation is independent of the choice and of the number
of independent variables. To find a relation between the second
total differentials, we need merely apply the general theorem to the
equation dF = 0, considered as an equation between u, v, w, du,
dv, dw, and so forth. The differentials of higher order than the
first of those variables which are chosen for independent variables
must, of course, be replaced by zeros.
Let us apply this theorem to calculate the successive total differ
entials of the implicit function defined by the equation (6), where
x and y are regarded as the independent variables. We find
*F i ^F j ^3F j
— dx + 7— a;/ + 7— dz = 0,
ox cy cz
dF dF 8F V2> , dF n
•T- dx + -z- dy + -r- dz ) + — <P* = 0,
dx dy dz / cz
and the first two of these equations may be used instead of the five
equations (7) and (8) ; from the expression for dz we may find the
two first derivatives, from that for d^z the three of the second order,
etc. Consider for example, the equation
Ax2 + A'y* + A"z* = l,
which gives, after two differentiations,
Ax dx + A 'ydy + A "z dz = 0,
A dx2 + A 'dy2 + A "dz2 + A " zd*z = 0,
whence
Axdx + A'ydy,
dg~ -- TTi - '
A"z
and, introducing this value of dz in the second equation, we find
A (A x* + A "z2} dx* + 2AA 'xy dx dy + A '(A'y* + A "z2} dy*
44 FUNCTIONAL RELATIONS [II, §24
Using Monge's notation, we have then
Ax A'y
p= ~IV q~ ~IV
A(Ax* + A"z*) _ AA'xy
~ "**'
This method is evidently general, whatever be the number of the
independent variables or the order of the partial derivatives which
it is desired to calculate.
Example. Let z = /(x, y) be a function of x and y. Let us try to calculate
the differentials of the first and second orders dx and d'2x, regarding y and z as
the independent variables, and x as an implicit function of them. First of all,
we have
dz = — dx + — dy.
dx dy
Since y and z are now the independent variables, we must set
d*y = d2z = 0,
and consequently a second differentiation gives
0 = ^dx» + 2 ^- dxdy + ?^-dy* + ^d?x.
£x2 dxdy dy* dx
In Monge's notation, using p, q, r, s, t for the derivatives of /(x, y), these
equations may be written in the form
dz — p dx + q dy,
0 = r dx2 + 2 s dx dy + tdy* + p d2x.
From the first we find
, dz — q dy
dx= - — -,
P
and, substituting this value of dx in the second equation,
rdz* + 2(ps-qr)dydz + (q*r -2pqs
<Px= -
The first and second partial derivatives of x, regarded as a function of y and
z, therefore, have the following values :
dx _ 1 8x _ q
dz p dy p
d*x _ r dzx _qr — ps d2x _ 2pqs — pH — q*r
dz2 p8 dy dz p3 dy2 ps
As an application of these formulae, let us find all those functions /(x, y)
which satisfy the equation
= 2pqs.
If, in the equation z =/(x, y), x be considered as a function of the two inde
pendent variables y and z, the given equation reduces to Xyt = 0. This means
II, § 25]
IMPLICIT FUNCTIONS
45
that xv is independent of y ; and hence xv = 0(z), where <f>(z) is an arbitrary
function of z. This, in turn, may be written in the form
which shows that x - y <f>(z) is independent of y. Hence we may write
where ^ (z) is another arbitrary function of z. It is clear, therefore, that all the
functions z =/(x, y) which satisfy the given equation, except those for which fx
vanishes, are found by solving this last equation for z. This equation represents
a surface generated by a straight line which is always parallel to the xy plane.
25. The general theorem. Let us consider a system of n equations
(E)
' ' '} xpi ul> **ll " 'i
* (x x • • • x ' u u •••w) = 0
between the n-\-p variables ui} u3, •••, un; xl} xy, •••, xp. Suppose
that these equations are satisfied for the values xv — x\, • • • , xp = xp,
u± = wj, • • • j un = u°n ; that the functions Fi are continuous and possess
first partial derivatives which are continuous, in the neighborhood of
this system of values; and, finally, that the determinant
du
does not vanish for
x,- —
uk =
Under these conditions there exists one and only one system of con
tinuous functions u^ = <f>i(xi, x2, •• •, x^), •• •, un — <}>n(x1, x2, • ••, xp~)
which satisfy the equations (E) and which reduce to u\, u\, •••, u°n,
for x, = x\, •••, xp = x*p*
*In his paper quoted above (ftn., p. 35) Goursat proves that the same conclusion
may be reached without making any hypotheses whatever regarding the derivatives
cFi/dXj of the functions F{ with regard to the x's. Otherwise the hypotheses remain
exactly as stated above. It is to be noticed that the later theorems regarding the
existence of the derivatives of the functions 4> would not follow, however, without
some assumptions regarding dFf/dXj. The proof given is based on the following
46 FUNCTIONAL RELATIONS [II, §26
The determinant A is called the Jacobian,* or the Functional Deter
minant, of the n functions Fu F2, • • -, Fn with respect to the n vari
ables ul} it?, • • •, un. It is represented by the notation
D(Flf F2, ...,F,,)
We will begin by proving the theorem in the special case of a
system of two equations in three independent variables x, y, z and
two unknowns u and v.
(9) Fi(x, y, z, u, v) = 0,
(10) Fi(x,y,z,u,v) = Q.
These equations are satisfied, by hypothesis, for x = x0,y = y0,z = z0,
u = u0, v = v0 ; and the determinant
dF\ dFj _ dF\ dFt
du cv dv cu
does not vanish for this set of values. It follows that at least one
of the derivatives dF^/dv, dF2/dv does not vanish for these same
values. Suppose, for definiteness, that oFl/8u does not vanish.
According to the theorem proved above for a single equation, the
relation (9) defines a function v of the variables x, y, z, u,
v =f(x, y, *> «)>
which reduces to v0 for x = x0, y = y0, z = zw u = u0. Replacing v
in the equation (10) by this function, we obtain an equation between
x, y, z, and u,
$(«, y, z, u} = Ft[x, y, z, u, f(x, y, z, «)] = 0,
lemma: Let f\(x\,3kt,---,vp; MI, u2, ••• ,un), •••,/„(«!, x?, •••,xp; MI, u2, •••, un) be n
functions of the n + p variables X{ and u^, which, together with the n2 partial deriva
tives cfi/GUfr, are continuous near Xi — 0, xz = 0, ••• , xp = 0, HI = 0, •••, un = 0. If
the n functions f{ and the n2 derivatives dfi/^Uf. all vanish for this system of values,
then the n equations
«i=/i. «2=/2, •"' «•»=/»
admit one and only one system of solutions of the form
where 01( 02> •••> 0n a™ continuous functions of the p variables Xi, x2, •• •, xp which
all approach zero as the variables all approach zero. The lemma is proved by means of
a suite of functions u^ =fi(x1,xz , •••,xp\ u[m~l\ w^"'-1), •••, u^'^) (i = l, 2, •••, n),
where M^O) = 0. It is shown that the suite of functions u\m) thus denned approaches a
limiting function U{, which 1) satisfies the given equations, and 2) constitutes the only
solution. The passage from the lemma to the theorem consists in an easy transforma
tion of the equations (E) into a form similar to that of the lemma. — TRANS.
*JACOBI, Crelle's Journal, Vol. XXII.
II, §25] IMPLICIT FUNCTIONS 47
which is satisfied for x = x0, y — y0, z = zw u = u0. Now
^ t , .
du 8u dv du
and from equation (9),
du ov ou
whence, replacing df/du by this value in the expression for
WP nhfaun
we obtain
d® D(u, v)
~du ~ dF^
dv
It is evident that this derivative does not vanish for the values x0,
y<» zo> uo- Hence the equation <I> = 0 is satisfied when u is replaced
by a certain continuous function u = <£ (x, y, «), which is equal to
MO when x = x0, y = y0, z = z0 ; and, replacing u by <£ (x, y, z) in
f(x, y, z, ?/), we obtain for v also a certain continuous function.
The proposition is then proved for a system of two equations.
We can show, as in § 21, that these functions possess partial
derivatives of the first order. Keeping y and z constant, let us
give x an increment Ax, and let AM and Ay be the corresponding
increments of the functions u and v. The equations (9) and (10)
then give us the equations
'-£ + . + A. + .' + A,, + ." = 0
• + + A. + ,' + A, + ,-' = 0,
' ' '
where e, e', e", rj, rj', rj" approach zero with Aa-, A«, Av. It follows
that
, ^ + c ^ + 77" - ^ + «» V-2 + 77
A;/. \ da; / \ go 7 / V g» / \ Ox
8Fi , A/^« , ,\ /^i , £"VaF24-V
\- f. II — p t]
y / \ ou
When Ax approaches zero, AM and Av also approach zero ; and hence
e, e', e", 77, 77', 77" do so at the same time. The ratio Aw /Ax therefore
approaches a limit ; that is, u possesses a derivative with respect to x :
48 FUNCTIONAL RELATIONS [II, §26
dFl cF2 dFl dF2
du dx dv dv dx
dx dFl dF2 dFi dFz
du dv dv du
It follows in like manner that the ratio Av/Aa; approaches a finite
limit dv /dx, which is given by an analogous formula. Practically,
these derivatives may be calculated by means of the two equations
8 Ft dFj du dFl dv _
dx du dx dv dx
dF2 dF^du dF^dv
o "T" <~\ ~r\~ T ~ ~^~~ == " !
OX CU OX CV OX
and the partial derivatives with respect to y and z may be found in
a similar manner.
In order to prove the general theorem it will be sufficient to show
that if the proposition holds for a system of (n — 1) equations, it
will hold also for a system of n equations. Since, by hypothesis,
the functional determinant A does not vanish for the initial values
of the variables, at least one of the first minors corresponding to the
elements of the last row is different from zero for these same values.
Suppose, for definiteness, that it is the minor which corresponds to
dFn/dun which is not zero. This minor is precisely
D(Fl} F2, -••,Fn_1).
D(UI, «2, ..., «„_,) '
and, since the theorem is assumed to hold for a system of (n — 1)
equations, it is clear that we may obtain solutions of the first (n — 1)
of the equations (E) in the form
M1 = ^1(a?1, a;,, •••,*„; Mn), ••-, un_l = fa^fa, x2, • •-, xp; un),
where the functions <£,. are continuous. Then, replacing u^ •••, wn_,
by the functions ^1? •••,<£„_! in the last of equations (E), we obtain
a new equation for the determination of un,
ai '••>*,; un) = Fn (xlt «,,-.., xp ; ^t, <^2, • • ., 0B_,, MII) = 0.
It only remains for us to show that the derivative d<b/dun does
not vanish for the given set of values x\, x°2, • • -, x°p, <; for, if so, we
can solve this last equation in the form
«» = ^0*i, *a, •••, *p),
where $ is continuous. Then, substituting this value of un ir
<^i5 •••> <f>n-i> we would obtain certain continuous functions foi
II, §25] IMPLICIT FUNCTIONS 49
HI, u2, ••-, un_l also. In order to show that the derivative in ques
tion does not vanish, let us consider the equation
- . . .
dun dui dun dun-i ^un dun
The derivatives 8<j>}/8un, d<jj2/dun, •••, d*n_i/d«n are given by the
(n — 1) equations
(12)
7J ~7, r * " T o ~o i o — V}
n-1 tj *Pl i i ^-^n-1 ^yn-1 • g-^n-l r\ .
and we may consider the equations (11) and (12) as n linear equa
tions for d<f>!/dun, • ••, d<f>n-i/dun, d®/dun) from which we find
cun D (MU t<2, •••,«„- i) D (M!, ?<2, • • • , wn)
It follows that the derivative d®/dun does not vanish for the initial
values, and hence the general theorem is proved.
The successive derivatives of implicit functions defined by several
equations may be calculated in a manner analogous to that used in
the case of a single equation. When there are several independent
variables it is advantageous to form the total differentials, from
which the partial derivatives of the same order may be found.
Consider the case of two functions u and v of the three variables
x, y, z defined by the two equations
F(x, y, z, u, v)=0,
The total differentials of the first order du and dv are given by the
two equations
3F . , 0F _ . SF . . 8F , .dF
-%- dx + ^- dy + ^- dz + -5- du + -r- dv = 0,
dx dy dz du cv
d® d$ .. d& , . d® , , d® -, A
-z- dx + -^- du + -^- dz + -r— du + -5— c?v = 0.
Sec cy ^s du cv
Likewise, the second total differentials dzu and d*v are given by the
equations
50 FUNCTIONAL RELATIONS [II, §26
dF V2) dF dF
- dx + -.. + — dv) + G/-d*u+™ d*v = 0,
cv du co
2> d& d&
+ ?-*d*u + ~d*v = 0,
CU CV
and so forth. In the equations which give dnu and dnv the deter
minant of the coefficients of those differentials is equal for all vahies
of n to the Jacobian D(F, <£)/Z>(w, v), which, by hypothesis, does not
vanish.
26. Inversion. Let MI, «2, • • • , un be n functions of the n independent vari
ables xi, x2, • • •, £n, such that the Jacobian D(UI, «2, • • • , u,,)/D(xi, x2, • • •, xn)
does not vanish identically. The n equations
/i g\
( Un = 0n(X1( X2, • • -, Xn)
define, inversely, Xi, x2, • • • , xn as functions of u\, M2, •••,«„. For, taking any
system of values x?, x£, • • • , x£, for which the Jacobian does not vanish, and
denoting the corresponding values of MI, w2, • • •, Un by uj, w!j, • • •, M°, there
exists, according to the general theorem, a system of functions
which satisfy (13), and which take on the values x", x", • • -, x°, respectively,
when MI = wj, • • • , un = unn. These functions are called the inverses of the func
tions 0i, 02, ••-, 0n, and the process of actually determining them is called
an inversion.
In order to compute the derivatives of these inverse functions we need merely
apply the general rule. Thus, in the case of two functions
u=f(x, y), v = <f>(x,y)t
if we consider u and v as the independent variables and x and y as inverse
functions, we have the two equations
whence
, 8f, , Bf , , d<f> , 30 ,
du = — dx + — dy, dv = — - dx + — - dy.
dx cy dx cy
^0 j %f -i c0 . df
— ait av du H — - dv
•, dy dy , dx dx
dx = , dy = •
a/^0_?/a0 a/ a0 _ ^/ ^0
dx dy dy dx dx dy dy dx
We have then, finally, the formulae
50 _ d_f
dx dy dx dy
du 8/a0_a/c0 dv d_fdj>_d_fd_$
dx dy dy dx dx dy dy dx
II, §27] IMPLICIT FUNCTIONS 51
£0 8f
dx dy ex
eu ~" df £0 _ cf d<t> eB^
ex e^ ey ex ex ey ey ex
27. Tangents to skew curves. Let us consider a curve C repre
sented by the two equations
l*i(*,y,«)-0,
(14) <
JF,(a5,y,«) = 05
and let x0, T/O, 20 be the coordinates of a point M0 of this curve, such
that at least one of the three Jacobians
dF\dF^ _d_F\d_F\ dF1dFt_dF\ 8F* 8Fl gF2 dF_i dF±
dy dz dz dy ' vz dx dx dz ex dy dij dx
does not vanish when x, y, z are replaced by x0, %, zot respectively.
Suppose, for defmiteness, that D(Fl} Fj/D(y, z) is one which does
not vanish at the point Mn. Then the equations (14) may be solved
in the form
y = ^(x)) z = t(x)>
where $ and \j/ are continuous functions of x which reduce to y0 and
z0, respectively, when x = x0. The tangent to the curve C at the
point 3/o is therefore represented by the two equations
X-x0 = F-7/Q = Z -g0
1 ' ^'(x0) " f (x0)
where the derivatives <#>'(cc) and i//(ce) may be found from the two
equations
In these two equations let us set x = x0)y = y0,z = 2^, and replace
*'(*„) and ^o) by ( F - T/O) / (X - *„) and (Z - «0)/(X - x0),
respectively. The equations of the tangent then become
62 FUNCTIONAL RELATIONS [II, §28
or
X xa Y y0 Z — z0
^(y, *) o ^>(«, «) o fl(«,y
The geometrical interpretation of this result is very easy. The
two equations (14) represent, respectively, two surfaces Sj and S2, of
which C is the line of intersection. The equations (15) represent
the two tangent planes to these two surfaces at the point 1/0; and
the tangent to C is the intersection of these two planes.
The formulae become illusory when the three Jacobians above all
vanish at the point M0. In this case the two equations (15) reduce
to a single equation, and the surfaces Si and S2 are tangent at the
point A/0. The intersection of the two surfaces will then consist, in
general, as we shall see, of several distinct branches through the
point M0.
II. FUNCTIONAL DETERMINANTS
28. Fundamental property. We have just seen what an important
role functional determinants play in the theory of implicit functions.
All the above demonstrations expressly presuppose that a certain
Jacobian does not vanish for the assumed set of initial values.
Omitting the case in which the Jacobian vanishes only for certain
particular values of the variables, we shall proceed to examine the
very important case in which the Jacobian vanishes identically.
The following theorem is fundamental.
Let HI, u2, •• • , un be n functions of the n independent variables
x\i xii '") xn" In order that there exist between these n functions
a relation II (MI} w2> • • •> un) == 0, which does not involve explicitly any
of the variables xly xz, • • • , xn, it is necessary and sufficient that the
functional determinant
D(UI, M2, •••, ?y)
should vanish identically.
In th.3 first place this condition is necessary. For, if such a rela
tion TL(UI, w2, • • •, wn) = 0 exists between the n functions HI, u%, • • •, un,
the following n equations, deduced by differentiating with respect to
each of the z's in order, must hold :
II, £28.1 FUNCTIONAL DETERMINANTS 53
endUl 8udu2 an 8un __
7; ^ — ~T~ "^ "o ~r ' ' ' T Q /-, — " ,
jfi , jf2 , » = Q.
dui 8xn du2 dxn dun dxn
and, since we cannot have, at the same time,
^5 = — = = ^5 = 0
£«! <7U2 CUH
since the relation considered would in that case reduce to a trivial
identity, it is clear that the determinant of the coefficients, which is
precisely the Jacobian of the theorem, must vanish.*
The condition is also sufficient. To prove this, we shall make
use of certain facts which follow immediately from the general
theorems.
1) Let u, v, w be three functions of the three independent variables
x, y, z, such that the functional determinant D(u, v, w)/D(x, y, z)
is not zero. Then no relation of the form
A du + /u, dv + v dw = 0
can exist between the total differentials du, dv, dw, except for
X = p, = v = 0. For, equating the coefficients of dx, dy, dz in the
foregoing equation to zero, there result three equations for X, p., v
which have no other solutions than X = /u, = v = 0.
2) Let w, u, v, w be four functions of the three independent
variables x, y, z, such that the determinant D (u, v, w} / D (x, y, s)
is not zero. We can then express x, y, z inversely as functions of
u, v, wt and substituting these values for x, y, z in o>, we obtain
a function
a, = $ (u, v, w)
of the three variables u, v, ^v. If by any process ivhatever we can
obtain a relation of the form
(16) du = P du + Q dv + R dw
*As Professor Osgood has pointed out, the reasoning here supposes that the
partial derivatives an / Si/i , dU /£«„, • • • , ^H / dUn do not all vanish simultaneously
for any system of values which cause U (ulf u2, ••-,«„) to vanish. This supposition
is certainly justified when the relation II = 0 is solved for one of the variables ut.
54 FUNCTIONAL RELATIONS [II, §28
between the total differentials dw, du, dv, dw, taken with respect to the
independent variables x, y, z, then the coefficients P, Q, R are equal,
respectively, to the three first partial derivatives of <£ (u, v, w) :
d& d$ 8<b
P = o ' Q = ~o ' •** = o '
Cu cv ow
For, by the rule for the total differential of a composite function
(§ 16), we have
d& d<b d&
«<D = - — du + -^— dv -|- — dw :
du cv cw
and there cannot exist any other relation of the form (16) between
d<a, du, dv, dw, for that would lead to a relation of the form
A. du + p. do + v dw = 0,
where X, /t, v do not all vanish. We have just seen that this is
impossible.
It is clear that these remarks apply to the general case of any
number of independent variables.
Let us then consider, for definiteness, a system of four functions
of four independent variables
(17)
X = Fl(x,y,z, *),
Y=Fi (x, y, z, t),
Z = F3(x, y, z, t),
T=Ft(x,y,z, t),
where the Jacobian D(Fl} F2, F3, Fi)/D(x, y, z, t) is identically
zero by hypothesis ; and let us suppose, first, that one of the first
minors, say D(F^ F2, Fs)/D(x, y, z), is not zero. We may then
think of the first three of equations (17) as solved for x, y, z as
functions of X, Y, Z, t ; and, substituting these values for x, y, z in
the last of equations (17), we obtain T as a function of A', Y, Z, t:
(18) T=*(X,Y,Z,t).
We proceed to show that this function $ does not contain the vari
able t, that is, that 8$ /dt vanishes identically. For this purpose
let us consider the determinant
II,
FUNCTIONAL DETERMINANTS
55
A =
QFi
dj\
df\
dX
dx
dt/
dz
dx
dF,
dFz
dz
dY
~dx~
dFs
ty
dFs
dz
dZ
dx
dF\
dz
dT
If, in this determinant, dX, dY, dZ, dT be replaced by their values
ox
tiy
Ct
and if the determinant be developed in terms of dx, dy, dz, dt, it turns
out that the coefficients of these four differentials are each zero ; the
first three being determinants with two identical columns, while the
last is precisely the functional determinant. Hence A = 0. But if
we develop this determinant with respect to the elements of the last
column, the coefficient of dTis not zero, and we obtain a relation of
the form
dT = P dX + Q dY + R dZ.
By the remark made above, the coefficient of dt in the right-hand
side is equal to d<i?/dt. But this right-hand side does not contain
dt, hence d&/dt = 0. It follows that the relation (18) is of the form
which proves the theorem stated.
It can be shown that there exists no other relation, distinct from
that just found, between the four functions X, Y, Z, T, independent
of x, y, z, t. For, if one existed, and if we replaced T by $>(X, Y, Z)
in it, we would obtain a relation between X, Y, Z of the form
U(X, Y, Z)=0, which is a contradiction of the hypothesis that
D(X, Y, Z)/D(x, y, z) does not vanish.
Let us now pass to the case in which all the first minors of the
Jacobian vanish identically, but where at least one of the second
minors, say D(Flt F^)/D(x, y}, is not zero. Then the first two of
equations (17) may be solved for x and y as functions of X. Y, z, t,
and the last two become
Z = *! (X, Y, z, t), T = ®.2 (A', Y, z, t).
56 FUNCTIONAL RELATIONS [n,
On the other hand we can show, as before, that the determinant
dX
dY
dZ
ex
ex
fy
¥
ex
dy
vanishes identically ; and, developing it with respect to the elements
of the last column, we find a relation of the form
dZ = FdX + QdY,
whence it follows that
In like manner it can be shown that
!r=0' '
dt
= 0;
and there exist in this case two distinct relations between the four
functions X, Y, Z, T, of the form
There exists, however, no third relation distinct from these two;
for, if there were, we could find a relation between X and Y, which
would be in contradiction with the hypothesis that D(X, Y} / D(x, y)
is not zero.
Finally, if all the second minors of the Jacobian are zeros, but
not all four functions X, Z, Y, T are constants, three of them are
functions of the fourth. The above reasoning is evidently general.
If the Jacobian of the n functions F1} F2, •• •, FH of the n independ
ent variables xly x2, •••, xn, together with all its (n — r + 1) -rowed
minors, vanishes identically, but at least one of the (n — r)- rowed
minors is not zero, there exist precisely r distinct relations between
the n functions ; and certain r of them can be expressed in terms
of the remaining (n — r), between which there exists no relation.
The proof of the following proposition, which is similar to the
above demonstration, will be left to the reader. The necessary and
sufficient condition that n functions of n + p independent variables be
connected by a relation which does not involve these variables is that
every one of the Jacobians of these n functions, with respect to any n
II, §28] FUNCTIONAL DETERMINANTS 57
of the independent variables, should vanish identically. In par
ticular, the necessary and sufficient condition that two functions
F i(#i , xz, •••, CCB) and F2(xl, xz,---, #„) should be functions of each
other is that the corresponding partial derivatives dF1/dxi and
dF2/dXf should be proportional.
Note. The functions F19 F2, •• •, Fn in the foregoing theorems may
involve certain other variables y1} y2) ••-, ym, besides xl, x2, •••, xn.
If the Jacobian D(Fl} Fz, •••, Fn)/D(xl, x2, ••-, ojn) is zero, the
functions JF\, F2, •••, Fn are connected by one or more relations
which do not involve explicitly the variables x1} x2, •••, xn, but
which may involve the other variables y1} y2, •••, ym.
Applications. The preceding theorem is of great importance. The funda
mental property of the logarithm, for instance, can be demonstrated by means
of it, without using the arithmetic definition of the logarithm. For it is proved
at the beginning of the Integral Calculus that there exists a function which is
defined for all positive values of the variable, which is zero when x — 1, and
whose derivative is l/x. Let/(x) be this function, and let
u=f(x)+f(y), v = xy.
Then
D (u, v) _
D (x, y)
x y =0.
y x
Hence there exists a relation of the form
f(x)+f(V) =
and to determine 0 we need only set y = 1, which gives f(x) = <j> (x). Hence,
since x is arbitrary,
f(z)+f(y)=f(xy).
It is clear that the preceding definition might have led to the discovery of
the fundamental properties of the logarithm had they not been known before the
Integral Calculus.
As another application let us consider a system of n equations in n unknowns
(MI, «2,
(19)
. Fn(Ul, W2,
where J/i, JT2) •••) Hn are constants or functions of certain other variables
*i» *2» •••» Xmi which may also occur in the functions .F,-. If the Jacobian
Z)(Fi, F2, • • •, Fn)/D(u\, «2i • • •> un) vanishes identically, there exist between
the n functions F, a certain number, say n — fc, of distinct relations of the form
• • • , Ft)t • • • , Flt = Un-k (F!, • • • , Fk).
58
FUNCTIONAL RELATIONS
[II, § 29
In order that the equations (19) be compatible, it is evidently necessary that
Ht + l = Hi (Hn .••,Hk),..-,Hn = UH-t (Hi, • • • , Hk),
and, if this be true, the n equations (19) reduce to k distinct equations. We
have then the same cases as in the discussion of a system of linear equations.
29. Another property of the Jacobian. The Jacobian of a system of n
functions of n variables possesses properties analogous to those of
the derivative of a function of a single variable. Thus the preceding
theorem may be regarded as a generalization of the theorem of § 8.
The formula for the derivative of a function of a function may be
extended to Jacobians. Let Flf F2, • • •, Fn be a system of n func
tions of the variables MI} u2, •••, un, and let us suppose that u^ w2>
• •-, ua themselves are functions of the n independent variables xlf
x • • • x. Then the formula
D(F
l,
-, Fn) D(UI,
D(xlt
D(x1}
follows at once from the rule for the multiplication of determinants
and the formula for the derivative of a composite function. For,
let us write down the two functional determinants
cj\
ou
dF
du
dxn dxu
cx
where the rows and the columns in the second have been inter
changed. The first element of the product is equal to
,
dFl
i
du,,
that is, to
?!, and similarly for the other elements.
30. Hessians. Let/(x, ?/, z) be a function of the three variables x, y, z. Then
the functional determinant of the three first partial derivatives cf/dx, Sf/cy,
df/dz,
a2/ a2/ a2/
ax2 ex 5y dx az
a2/ a2/ a2/
ax cy a?/2
a2/ a2/
ex cz cy oz
dydz
a2/
cz-
II, § 30]
FUNCTIONAL DETERMINA NTS
59
is called the Hessian of f(x, y, z). The Hessian of a function of n variables is
defined in like manner, and plays a role analogous to that of the second deriva
tive of a function of a single variable. We proceed to prove a remarkable
invariant property of this determinant. Let us suppose the independent vari
ables transformed by the linear substitution
(X= aX+ [3Y+ yZ,
y= a'X + p'Y+ y'Z,
(19')
where X, F, Z are the transformed variables, and or, 0, 7, • • • , 7" are constants
such that the determinant of the substitution,
a J8 7
A = a' /3' 7'
a" /3" 7"
is not zero. This substitution carries the function /(x, y, z) over into a new
function F(X, Y, Z) of the three variables X, Y, Z. Let II (X, F, Z) be the
Hessian of this new function. We shall show that we have identically
II (X, F, Z) = A2/t(x, ?/, z),
where x, ?/, z are supposed replaced in /i(x, y, z) by their expressions from (19').
For we have
fZF dF dF^
H =
7 £^ ?I\
BY' cZ ) ~\dX' aT' aZ/ D(x, y,
D(X, Y, Z) D(x, y, z) D(X, Y, Z)
and if we consider cf/cx, cf/dy, df/dz, for a moment, as auxiliary variables,
we may write
By cz D(x, y,
^, ^, Kl
cx dy dz /
D(x,y,z) U(X, Y, Z)
But from the relation F(X, Y, Z) =f(x, y, z), we find
dF cf ,cf ,,Bf
-— = a — + a' — + a" — ,
dX ex cy dz
dY
dy
whence
3F a/ , a/ , „ c/
-^ = 7 — + 7 — + 7 — »
c:Z cx cy cz
d_F dF
er' ez
and hence, finally,
dx dy dz
a a' a"
7 7
= A;
H=
D(x, y, z)
D(X, Y, Z)
It is clear that this theorem is general.
'- = A2/i-
60 FUNCTIONAL RELATIONS [II, §30
Let us now consider an application of this property of the Hessian. Let
/(x, y) = ox3 + 3 bx*y + 3 cxy2 + dy*
be a given binary cubic form whose coefficients a, b, c, d are any constants.
Then, neglecting a numerical factor,
h =
ax + by bx 4 cy
bx + cy ex + dy
= (ac - &2)x2 + (ad - bc)xy + (bd - c2)y2,
and the Hessian is seen to be a binary quadratic form. First, discarding the
case in which the Hessian is a perfect square, we may write it as the product of
two linear factors :
h = (mx + ny) (px + qy).
If, now, we perform the linear substitution
mx + ny = X, px + qy = Y,
the form/(x, y) goes over into a new form,
F(X, Y) = AX* + 3 BX2 Y+3 CXY2 + DY8,
whose Hessian is
H(X, Y) = (AC - B2) X2 + (AD - EC] XY + (BD - C2) F2,
and this must reduce, by the invariant property proved above, to a product of
the form KXY. Hence the coefficients A, B, C, D must satisfy the relations
If one of the two coefficients B, C be different from zero, the other must be so,
and we shall have
-?• -f-
F(X, Y) = — (B*X* + 3 B2 CX* Y + 3 BC* XY2 + C* Y3) = (B^+^Y)\
whence F(X, Y), and hence /(x, y), will be a perfect cube. Discarding this
particular case, it is evident that we shall have B = C = 0 ; and the polynomial
F(X, Y) will be of the canonical form
AX* + DY3.
Hence the reduction of the form /(x, y) to its canonical form only involves the
solution of an equation of the second degree, obtained by equating the Hessian
of the given form to zero. The canonical variables X, Y are precisely the two
factors of the Hessian.
It is easy to see, in like manner, that the form/(x, y) is reducible to the form
AX3 + BX2 Y when the Hessian is a perfect square. When the Hessian van
ishes identically /(x, y) is a perfect cube :
/(x, y) = (ax
II, §31] TRANSFORMATIONS 61
III. TRANSFORMATIONS
It often happens, in many problems which arise in Mathematical
Analysis, that we are led to change the independent variables. It
therefore becomes necessary to be able to express the derivatives
with respect to the old variables in terms of the derivatives with
respect to the new variables. We have already considered a problem
of this kind in the case of inversion. Let us now consider the
question from a general point of view, and treat those problems
which occur most frequently.
31. Problem I. Let y be a function of the independent variable x,
and let t be a new independent variable connected luith x by the relation
x = <£(£). It is required to express the successive derivatives of y with
respect to x in terms of t and the successive derivatives of y with
respect to t.
Let y=f(x) be the given function, and F(t) =/[<£(£)] the func
tion obtained by replacing x by <j>(t) in the given function. By the
rule for the derivative of a function of a function, we find
dy dy ,. .
•37 = ~r~ X 9 m,
at ax
whence
dy
dt yt
This result may be stated as follows : To find the derivative of y
with respect to x, take the derivative of that function with respect to t
and divide it by the derivative of x with respect to t.
The second derivative d2y/dx* may be found by applying this
rule to the expression just found for the first derivative. We find :
-I
dLl = ±_ -y^'(0-y^"(0.
dx* w) ' [>'(0]'
and another application of the same rule gives the third derivative
62 FUNCTIONAL RELATIONS [H, §32
or, performing the operations indicated,
_
<*»• [>'(OJ6
The remaining derivatives may be calculated in succession by
repeated applications of the same rule. In general, the nth deriva
tive of y with respect to x may be expressed in terms of <}>'(£), <j>"(t),
• ••, <£(n)(£), and the first n successive derivatives of y with respect to
t. These formulae may be arranged in more symmetrical form.
Denoting the successive differentials of x and y with respect to t by
dx, dy, dzx, d*y, ••-, d"x, dny, and the successive derivatives of y
with respect to x by y', y", •••, y(n\ we may write the preceding
formulae in the form
(20)
y — 7
9 . dx
f _ dx d2y — dy 6?
dx3
x2 - 3 d?y dx dzx + 3dy (d*x)* - dy d*x dx
y ~~ 5
The independent variable t, with respect to which the differentials
on the right-hand sides of these formulae are formed, is entirely
arbitrary ; and we pass from one derivative to the next by the
recurrent formula
, ,
«<»>=
the second member being regarded as the quotient of two differen
tials.
32. Applications. These formulas are used in the study of plane
curves, when the coordinates of a point of the curve are expressed in
terms of an auxiliary variable t.
« =/(*)» y = * co
in order to study this curve in the neighborhood of one of its points
it is necessary to calculate the successive derivatives y', ?/", • • • of y
with respect to x at the given point. But the preceding formulas
give us precisely these derivatives, expressed in terms of the succes
sive derivatives of the functions f(t) and <j> (#), without the necessity
II, §32] TRANSFORMATIONS 63
of having recourse to the explicit expression of y as a function of x,
which it might be very difficult, practically, to obtain. Thus the
first formula
y> = dx = f'(t)
gives the slope of the tangent. The value of y" occurs in an impor
tant geometrical concept, the radius of curvature, which is given by
the formula
which we shall derive later. In order to find the value of R, when
the coordinates x and y are given as functions of a parameter t, we
need only replace y' and y" by the preceding expressions, and we
find
(dx2 4- dy^Y
R = . , — ^r~ "
where the second member contains only the first and second deriva
tives of x and y with respect to t.
The following interesting remark is taken from M. Bertrand's Traitt de
Calcul differentiel et integral (Vol. I, p. 170). Suppose that, in calculating some
geometrical concept allied to a given plane curve whose coordinates x and y are
supposed given in terms of a parameter £, we had obtained the expression
F(x, y, dx, dy, d2x, d2y, • • -, dnx, d»y),
where all the differentials are taken with respect to t. Since, by hypothesis,
this concept has a geometrical significance, its value cannot depend upon the
choice of the independent variable t. But, if we take x = t, we shall have
dx — dt, dzx = d3x = • • • = dax = 0, and the preceding expression becomes
f(x, y, y', y", ••••> 2/('°) ;
which is the same as the expression we would have obtained by supposing at the
start that the equation of the given curve was solved with respect to y in the
form y = *(«). To return from this particular case to the case where the inde
pendent variable is arbitrary, we need only replace y', y", • • • by their values
from the formulae (20). Performing this substitution in
we should get back to the expression F(x, y, dx, dy, d'2x, d2y, • • •) with which
we started. If we do not, we can assert that the result obtained is incorrect.
For example, the expression
dxd'2y + dyd2x
64 FUNCTIONAL RELATIONS [II, §33
cannot have any geometrical significance for a plane curve which is independent
of the choice of the independent variable. For, if we set x = t, this expression
reduces to y" /(I + y'2)$ ; and, replacing y' and y" by their values from (20), we
do not get back to the preceding expression.
33. The formulae (20) are also used frequently in the study of
differential equations. Suppose, for example, that we wished to
determine all the functions y of the independent variable x, which
satisfy the equation
(21) (1_^£*_e eg + „.„ = „,
where n is a constant. Let us introduce a new independent variable
t, where x = cos t. Then we have
dy
dy dt
dx — sin t
d*y dy
smt-jfi.— cost —
d?y at* dt <
dx2 sin8 1
and the equation (21) becomes, after the substitution,
(22)
It is easy to find all the functions of t which satisfy this equation,
for it may be written, after multiplication by 2 dy /dt,
whence
where a is an arbitrary constant. Consequently
or
— 71 = 0.
II, § 34] TRANSFORMATIONS 65
The left-hand side is the derivative of arc sin (y/a) — nt. It follows
that this difference must be another arbitrary constant b, whence
y = a sm(nt + &),
which may also be written in the form
y = A sin nt + B cos nt.
Returning to the original variable x, we see that all the functions of
x which satisfy the given equation (21) are given by the formula
y = A sin (n arc cos a) + B cos (n arc cos a),
where A and B are two arbitrary constants.
34. Problem II. To every relation between x and y there corresponds,
by means of the transformation x = f(t, u), y = <f>(t, u*), a relation
between t and u. It is required to express the derivatives of y with
respect to x in terms of t, u. and the derivatives of u with respect to t.
This problem is seen to depend upon the preceding when it is
noticed that the formulae of transformation,
give us the expressions for the original variables x and y as func
tions of the* variable t , if we imagine that u has been replaced in
these formulas by its value as a function of t. We need merely
apply the general method, therefore, always regarding x and y as
composite functions of t, and u as an auxiliary function of t. We
find then, first,
8<jt 8<ft du
dy _dy dx dt du dt
dx dt ' dt df df du
dt du dt
and then
d?y __ d (dy\ dx
dx"2 dt \dxj ' dt
or, performing the operations indicated,
, _ _ , ,
SuBtdt du*\dt) gu dfr* \dt + du dt/\dt*
£t du dt
66 FUNCTIONAL RELATIONS [II, §33
In general, the nth derivative y(n) is expressible in terms of t, u, and
the derivatives du/dt, d?vi/dt2, •••, dnu/dtn.
Suppose, for instance, that the equation of a curve be given in
polar coordinates p = /(o>). The formulae for the rectangular coor
dinates of a point are then the following :
x = p cos <D, y — p sin w.
Let p', p", • • • be the successive derivatives of p with respect to w,
considered as the independent variable. From the preceding formulae
we find
dx = cos (a dp — p sin w e?w,
dy = sin o> dp + p cos w d<a,
d2x = cosu) dzp — 2 sin w dai dp — p cos w da?,
d2y — sinw d2p + 2 cosw rfw ffy — p sin w «7(o2,
whence
<&e2 + dif1 = dp2 + p2 rfw2,
— dij d^x = 2 du dp2 — p d<a d2p + p2 c?w3.
The expression found above for the radius of curvature becomes
p' + *pm-pp
35. Transformations of plane curves. Let us suppose that to every
point m of a plane we make another point M of the same plane cor
respond by some known construction. If we denote the coordinates
of the point m by (x, y) and those of M by (X, F), there will exist,
in general, two relations between these coordinates of the form
(23) X=f(x,y), Y=4>(x, y}.
These formulae define a point transformation of which numerous
examples arise in Geometry, such as projective transformations, the
transformation of reciprocal radii, etc. When the point m describes
a curve c, the corresponding point M describes another curve C, whose
properties may be deduced from those of the curve c and from the
nature of the transformation employed. Let y', ?/", • • • be the suc
cessive derivatives of y with respect to x, and F', F", • • • the succes
sive derivatives of F with respect to X. To study the curve C it
is necessary to be able to express F', F", ••• in terms of x, y, y',
y", ••-. This is precisely the problem which we have just discussed ;
and we find
II, § 36]
TRANSFORMATIONS
67
dY
Y'
dx
dx dy '
Y"
dX
dx
dY'
dx
dx dy
dX
dx
/df df ,V
1 o i — ~ y }
\dx dy I
and so forth. It is seen that Y' depends only on x, y, y'. Hence,
if the transformation (23) be applied to two curves c, c', which are
tangent at the point (x, ?/), the transformed curves C, C' will also
be tangent at the corresponding point (A', F). This remark enables
us to replace the curve c by any other curve which is tangent to it
in questions which involve only the tangent to the transformed
curve C.
Let us consider, for example, the transformation defined by the
formulae
Y =
<-— A
x2 + y'2
which is the transformation of reciprocal radii, or inversion, with
the origin as pole. Let m be a point of a curve c and M the cor
responding point of the curve C. In
order to find the tangent to this curve
C we need only apply the result of
ordinary Geometry, that an inversion
carries a straight line into a circle
through the pole.
Let us replace the curve c by its
tangent mt. The inverse of mt is a
circle through the two points Mand O,
whose center lies on the perpendicular
Ot let fall from the origin upon mt. The tangent MT to this circle
is perpendicular to AM, and the angles Mmt and mMT are equal,
since each is the complement of the angle mOt. The tangents mt
and MT are therefore antiparallel with respect to the radius vector.
36. Contact transformations. The preceding transformations are
not the most general transformations which carry two tangent
curves into two other tangent curves. Let us suppose that a point
M is determined from each point m of a curve c by a construction
FIG. 5
68 FUNCTIONAL RELATIONS [II, §36
which depends not only upon the point m, but also upon the tangent
to the curve c at this point. The formulae which define the trans
formation are then of the form
(24) X = /(*, y, y-), Y=<j>(x, y, y1) ;
and the slope Y' of the tangent to the transformed curve is given
by the formula
dx dy dy> y
In general, F' depends on the four variables x, y, y', y" ; and if we
apply the transformation (24) to two carves c, c' which are tangent
at a point (x, y~), the transformed curves C, C' will have a point
(X, Y) in common, but they will not be tangent, in general, unless
y" happens to have the same value for each of the curves c and c'.
In order that the two curves C and C' should always be tangent, it
is necessary and sufficient that Y' should not depend on y"; that is,
that the two functions f(x, y, y') and <£ (x, y, y') should satisfy the
condition
In case this condition is satisfied, the transformation is called a
contact transformation. It is clear that a point transformation is a
particular case of a contact transformation.*
Let us consider, for example, Legendre's transformation, in which
the point M, which corresponds to a point (x, y) of a curve c, is given
by the equations
X = y', Y=xy'-y;
from which we find
Y, _dY _xjf _
~ dX - y"
which shows that the transformation is a contact transformation.
In like manner we find
dY' dx 1
V11' =r
dX y"dx y"
y'
dX
*Legendre and Ampere gave many examples of contact transformations. Sophus
Lie developed the general theory in various works ; see in particular his Geometric
der Beruhrungstransformationen. See also JACOBI, Vorlesungen iiber Dynamik.
II, §37] TRANSFORMATIONS 69
and so forth. From the preceding formulae it follows that
x = Y', y = XY'-Y, y' = X,
which shows that the transformation is involutory.* All these prop
erties are explained by the remark that the point whose coordinates
are X = y', Y = xy' — y is the pole of the tangent to the curve c at
the point (x, y) with respect to the parabola x2 — 2 y = 0. But, in
general, if M denote the pole of the tangent at m to a curve c with
respect to a directing conic 2, then the locus of the point M is a,
curve C whose tangent at M is precisely the polar of the point m
with respect to 2. The relation between the two curves c and C is
therefore a reciprocal one ; and, further, if we replace the curve c by
another curve c', tangent to c at the point m, the reciprocal curve C'
will be tangent to the curve C at the point M.
Pedal curves. If, from a fixed point O in the plane of a curve c, a perpen
dicular OM be let fall upon the tangent to the curve at the point m, the locus of
the foot M of this perpendicular is a curve (7, which is called the pedal of the
given curve. It would be easy to obtain, by a direct calculation, the coordinates
of the point Jlf, and to show that the trans
formation thus defined is a contact transfor
mation, but it is simpler to proceed as follows.
Let us consider a circle 7 of radius E, de
scribed about the point 0 as center; and let ?MI
be a point on OM such that Om\ x OM= E2.
The point mi is the pole of the tangent mt
with respect to the circle ; and hence the
transformation which carries c into C is the
result of a transformation of reciprocal po-
lars, followed by an inversion. When the
point m describes the curve c, the point mi,
the pole of mt, describes a curve Ci tangent
to the polar of the point m with respect to
the circle 7, that is, tangent to the straight line miti, a perpendicular let fall
from mi upon Om. The tangent M Tto the curve C and the tangent m\ti to the
curve Ci make equal angles with the radius vector OmiM. Hence, if we draw
the normal MA, the angles AMO and AOM are equal, since they are the comple
ments of equal angles, and the point A is the middle point of the line Om. It
follows that the normal to the pedal is found by joining the point Mto the center
of the line Om.
37. Projective transformations. Every function y which satisfies the equation
y" = 0 is a linear function of x, and conversely. But, if we subject x and y to
the projective transformation
* That is, two successive applications of the transformation lead us back to the
original coordinates. — TRANS.
70 FUNCTIONAL RELATIONS [II, §38
_ aX + bY+c _ a'X+ b' Y + c'
a" X + b" Y + c" a" X -f b" Y + c"
a straight line goes over into a straight line. Hence the equation y" = 0 should
become d'^Y/dX"2 — 0. In order to verify this we will first remark that the
general projective transformation may be resolved into a sequence of particular
transformations of simple form. If the two coefficients a" and b" are not both
zero, we will set X\ = a" X -\- b" Y + c" ; and since we cannot have at the same
time ab" — ba" = 0 and a' b" — b' a" = 0, we will also set YI — a' X + b' Y + c',
on the supposition that a' b" — b' a" is not zero. The preceding formulae may
then be written, replacing X and Y by their values in terms of Xi and Fl5 in
the form
YI a Xi + /3 FI + 7 YI 7
A! Xi AI Xi
It follows that the general projective transformation can be reduced to a
succession of integral transformations of the form
x — aX + bY + c, y - a' X + b' Y + c',
combined with the particular transformation
1 Y
x = — , y = — .
X X
Performing this latter transformation, we find
and
dy _-.-_
~ dx ~ '~~
y" = ~y~ = - XY"(- X*) = Xs Y".
dx
Likewise, performing an integral projective transformation, we have
d y a' + b' Y'
y -
a + bY'
„ _ dt/ _ (ab'-ba')Y"
~ dx (a + bY')3
In each case the equation y" — 0 goes over into Y" = 0.
We shall now consider functions of several independent variables, and, for
definiteness, we shall give the argument for a function of two variables.
38. Problem III. Let w = f(x, y) be a function of the two independ
ent variables x and y, and let u and v be two new variables connected
with the old ones by the relations
It is required to express the partial derivatives of u with respect to the
variables x and y in terms of u, v, and the partial derivatives of u> with
respect to u and v.
II,
TRANSFORMATIONS 71
Let w = F(u, v) be the function which results from/(x, y) by the
substitution. Then the rule for the differentiation of composite
functions gives
c oj 8 a) 8 <J> d o» d\ff
cu dx cu dy du
Cd) C w C (jt d ta C\}/
dv dx cv dy dv
whence we may find d<a/dx and du/dy; for, if the determinant
D(<f>, \ji)/D(u, v) vanished, the change of variables performed
would have no meaning. Hence we obtain the equations
(25)
dw d(a d d>
o~ = A a — r B ~z~'
ex cu cv
Cu) __ Cd) Coi
— — — c -^ + u ~^~~>
cy cu cv
where A, B, C, D are determinate functions of u and v ; and these
formulae solve the problem for derivatives of the first order. They
show that the derivative of a function with respect to x is the sum of
the two products formed by multiplying the two derivatives with respect
to u and v by A and B, respectively. The derivative with respect to
y is obtained in like manner, using C and D instead of A and B,
respectively. In order to calculate the second derivatives we need
only apply to the first derivatives the rule expressed by the preced
ing formulae ; doing so, we find
£2<o d /8u>\ d I t d& i
575 ==2~\'"-~/==2\p — r •» •
C/X Q<Kf \ t/JC / GOT/ \ C/it
\ / \
(31 C o> v to \ .- w . _ _ _
= A — (A - + B „- + B y- (A r- + B -5-
cu\ Cu cv / dv\ cu
or, performing the operations indicated,
^= A[A-z£+Br-%:+-^-- + -z
^-^2 v ^"^ CU CV CU CU CU CV
-B t-^ +
«
My P cy </« 0v cv
and we could find 82(a/dxdy, 32<a/di/2 and the following derivatives
in like manner. In all differentiations which are to be carried out
we need only replace the operations d /dx and d /dy by the operations
'd 8 d d
A-Z-+B-Z-* C— + D^-,
du Cv cu cv
72
FUNCTIONAL RELATIONS
[II, §38
respectively. Hence everything depends upon the calculation of the
coefficients A, B, C, D.
Example I. Let us consider the equation
(26)
C CO , 0 (i3 dJ
a — +26 -- + c — = 0,
ex ci/ cy*
where the coefficients a, 6, c are constants ; and let us try to reduce this equa
tion to as simple a form as possible. We observe first that if a = c = 0, it would
be superfluous to try to simplify the equation. We may then suppose that c,
for example, does not vanish. Let us take two new independent variables u
and v, defined by the equations
u = x + ay, v = x + py,
where a and /3 are constants. Then we have
c u < <jj cu
cx
d<a
-
8y
du I w
a— + P — ,
du cv
and hence, in this case, A = B = 1, C = a, D = p. The general formulae then
give
dx?
~dii? cucv "ai?2"'
ducv
eu2
and the given equation becomes
(a + 26a + ca2)^ + 2 [a + b(a + ft) + ca/3]-^- + (a -(-
au-^ au a» CB
It remains to distinguish several cases.
First case. Let 62 — ac> 0. Taking for a and £ the two roots of the equation
a .)- 2 6r + cr2 = 0, the given equation takes the simple form
cudv
= 0.
Since this may be written
we see that dta/Su must be a function of the single variable, w, say/(w). Let
F(u) denote a function of u such that F'(u) =f(u). Then, since the derivative
of w — F(u) with respect to u is zero, this difference must be independent of w,
and, accordingly, u = F(u) + *(«). The converse is apparent. Returning to
the variables x and y, it follows that all the functions w which satisfy the equation
(26) are of the form
II, §38] TRANSFORMATIONS 73
where F and $ are arbitrary functions. For example, the general integral of
the equation
c2w Oc2w
— = a2 — ,
cy2 dx2
which occurs in the theory of the stretched string, is
w =f(x + ay) + <f> (x - ay).
Second case. Let b'1 — ac = 0. Taking a equal to the double root of the equa
tion a -f 26r -f cr2 = 0, and |3 some other number, the coefficient of d^w/dudv
becomes zero, for it is equal to a + ba + p(b + car). Hence the given equation
reduces to 52w/cz>2 = 0. It is evident that w must be a linear function of t>,
w = vf(u) + <f> (u), where f(u) and <f> (u) are arbitrary functions. Returning to
the variables x and y, the expression for w becomes
w = (x + Py)f(x + ay) + <f>(x + ay),
which may be written
w = [x + ay + (p - a)y]f(x + ay) + <f>(x + ay),
or. finally,
w = yF(x + ay) + <l>(x + ay).
Third case. If 62 — ac < 0, the preceding transformation cannot be applied
without the introduction of imaginary variables. The quantities a and /3 may
then be determined by the equations
a + 26a + c a2 =
a + b(a + p)
which give
2b 2&2-ac
a + /3= , a/3=
c c2
The equation of the second degree,
26 262-ac
r2 H r H = 0,
c c2
whose roots are a and £, has, in fact, real roots. The given equation then
becomes
a2w c2w
Aw = H = 0.
du2 c«2
This equation Aw = 0, which is known as Laplace's Equation, is of fundamental
importance in many branches of mathematics and mathematical physics.
Example II. Let us see what form the preceding equation assumes when we
set x = p cos <£, y = p sin 0. For the first derivatives we find
8 u du Su
— = — cos^ H
8p dx dy
i u ("ui d u
— = p sin <b -\ p cos <z>,
p *
74 FUNCTIONAL RELATIONS [H,§a9
or, solving for dw/dx and du/cy,
du du sin rf> du
- — COS <p i
dx dp p dip
du du cosrf) du
- =sm0— +
dy dp p dip
Hence
a I du sin0 du\ sind> 8 t
— ( costf> --- - — ) -- — (
dp\ dp p 8<f>/ p d<t>\
du simp du
--- -
dp p
d'2u sin20a2w 2 sin </> cos 0 S2w 2sin0cos<£dw sin20cw
= COS2 <t> -- 1 ------- 1 ---- 1 --- i
a/?2 p2 dtf>2 p dp dip p2 d<f> p dp
and the expression for d2u/dy2 is analogous to this. Adding the two, we find
39. Another method. The preceding method is the most practical
when the function whose partial derivatives are sought is unknown.
But in certain cases it is more advantageous to use the following
method.
Let z =f(x, y) be a function of the two independent variables x
and y. If x, y, and z are supposed expressed in terms of two aux
iliary variables u and v, the total differentials dx, dy, dz satisfy the
relation
O /> o /•
dz = -^- dx + -£- dy.
ex cy '
which is equivalent to the two distinct equations
_
du dx du dy du
dz _d_fdx
dv dx dv dy dv
whence df/dx and df/dy may be found as functions of u, v, dz/du,
dz/dv, as in the preceding method. But to find the succeeding
derivatives we will continue to apply the same rule. Thus, to find
d2f/dx2 and d2f/dxdy, we start with the identity
>
dx2 dxcy
which is equivalent to the two equations
d(dx) = d2fdx | ay dy}
du dx2 du dx dy du
II, §39] TRANSFORMATIONS 75
: | }
dec2 ov dx dy dv
where it is supposed that df/dx has been replaced by its value cal
culated above. Likewise, we should find the values of d2f/dx dy and
2 by starting with the identity
df\ a2/ 82f
a I — Q ?T *** ' ~cT~a *%•
dy/ dxdy dy2
The work may be checked by the fact that the two values of
c2f/8x dy found must agree. Derivatives of higher order may be
calculated in like manner.
Application to surfaces. The preceding method is used in the study
of surfaces. Suppose that the coordinates of a point of a surface S
are given as functions of two variable parameters u and v by means
of the formulae
(27) x=f(u,v), y = $(u,v), z = f(u,v).
The equation of the surface may be found by eliminating the vari
ables u and v between the three equations (27); but we may also
study the properties of the surface S directly from these equations
themselves, without carrying out the elimination, which might be
practically impossible. It should be noticed that the three Jacobians
D(«, -y) D(u, v)
cannot all vanish identically, for then the elimination of w and v
would lead to two distinct relations between x, y, z, and the point
whose coordinates are (x, y, %) would map out a curve, and not a sur
face. Let us suppose, for definiteness, that the first of these does not
vanish : D(f, <j>)/D(u, v) =£ 0. Then the first two of equations (27)
may be solved for u and v, and the substitution of these values in the
third would give the equation of the surface in the form z = F(x, y).
In order to study this surface in the neighborhood of a point we need
to know the partial derivatives p, q, r,s,t, ••• of this function F(x, y)
in terms of the parameters u and v. The first derivatives p and q
are given by the equation
dz = p dx -f- q dy,
which is equivalent to the two equations
76 FUNCTIONAL RELATIONS [II, §40
^ - n^4-n^
TT— = p T; -- r q r~
du f du du
Q a
cv cv cv
from which p and q may be found. The equation of the tangent
plane is found by substituting these values of p and q in the equation
Z - z = p(X - *) + q(Y - y),
and doing so we find the equation
The equations (28) have a geometrical meaning which is easily
remembered. They express the fact that the tangent plane to the
surface contains the tangents to those two curves on the surface which
are obtained by keeping v constant while u varies, and vice versa*
Having found p and q, p =/i(w, v), q = f2(u, v), we may proceed
to find r, s, t by means of the equations
dp = r dx + s dy,
\
dq = s dx + t dy,
each of which is equivalent to two equations ; and so forth.
40. Problem IV. To every relation between x, y, z there corresponds
by means of the equations
(30) x =/(w, v, w), y = <£ (M, v, w), z = \j/(u, v, w),
a new relation between u, v, w. It is required to express the partial
derivatives of z with respect to the variables x and y in terms of u, v, w,
and the partial derivatives of iv with respect to the variables u and v.
This problem can be made to depend upon the preceding. For,
if we suppose that w has been replaced in the formulae (30) by a
function of u and v, we have x, y, z expressed as functions of the
* The equation of the tangent plane may also be found directly. Every curve on
the surface is defined by a relation between u and w, say v = U (u) ; and the equations
of the tangent to this curve are
X-x Y-y Z-z
df df ~ dd> d4> ~ d4> d&
f- + -f n'(«) ^ + ~ n'(«) -^- + -— H'(M)
du dv du dv du dv
The elimination of IT(a) leads to the equation (29) of the tangent plane.
II, §41] TRANSFORMATIONS 77
two parameters u and v; and we need only follow the preceding
method, considering /, <£, ^ as composite functions of u and v, and
w as an auxiliary function of u and v. In order to calculate the
first derivatives p and y, for instance, we have the two equations
_ , ,
P a Q — T5 I cT~ i T~ ~o — 1+7 "^ -- P ^ -- o
* cu ow 8u ' du dw cu
_ a^ a_w
dv dw dv \d» 8w dv \8v + 8w
The succeeding derivatives may be calculated in a similar manner.
In geometrical language the above problem may be stated as fol
lows : To every point m of space, whose coordinates are (x, y, z),
there corresponds, by a given construction, another point M, whose
coordinates are X, Y, Z. When the point m maps out a surface S,
the point M maps out another surface 2, whose properties it is pro
posed to deduce from those of the given surface S.
The formulae which define the transformation are of the form
x =f(x> y> «), y = <t> (*, y, *),
Let
Y)
be the equations of the two surfaces S and 2, respectively. The
problem is to express the partial derivatives P, Q, R, S,T, ••• of the
function $(A", Y) in terms of x, y, z and the partial derivatives
p, q, r, s, t, • • • of the function F(x, y). But this is precisely the
above problem, except for the notation.
The first derivatives P and Q depend only on x, y, z, p, q ; and
hence the transformation carries tangent surfaces into tangent sur
faces. But this is not the most general transformation which enjoys
this property, as we shall see in the following example.
41. Legendre's transformation. Let z =f(x, y) be the equation of
a surface S, and let any point m (x, ?/, z) of this surface be carried
into a point M, whose coordinates are X, Y, Z, by the transformation
X=p, Y = q, Z — px + qy — z.
Let Z = $ (X, Y) be the equation of the surface 2 described by the
point M. If we imagine z, p, q replaced by /, df/dx, df/dy, respec
tively, we have the three coordinates of the point M expressed as
functions of the two independent variables x and y.
78 FUNCTIONAL RELATIONS [II, §41
Let P, Q, R, S, T denote the partial derivatives of the function
$>(X, Y). Then the relation
dZ = PdX+ QdY
becomes
p dx + q dy + x dp + y dq — dz = P dp + Q, dq,
or
x dp + y dq = P dp -f Q dq.
Let us suppose that p and q, for the surface S, are not functions of each
other, in which case there exists no identity of the form \dp + p.dq=. 0,
unless X = fi — 0. Then, from the preceding equation, it follows that
In order to find R, S, T we may start with the analogous relations
dP = RdX + SdY,
dQ = SdX+ TdY,
which, when X, Y, P, Q are replaced by their values, become
dx = R (r dx + s dy) + S (s dx + t dy) ,
dy = S (r dx + s dy~) + T(sdx + t dy} ;
whence
and consequently
t — s r
rt — s2 rt — s2 rt — s2
From the preceding formulae we find, conversely,
x = P, y=Q, z = PX+QY-Z, p = X, q = Y,
T - S R
t —
RT- ^ RT- .S'2 RT - S'2
which proves that the transformation is involutory. Moreover, it
is a contact transformation, since X, Y, Z, P, Q depend only on x,
y, z, p, q. These properties become self-explanatory, if we notice
that the formulae define a transformation of reciprocal polars with
respect to the paraboloid
x2 + y2 - 2 z = 0.
Note. The expressions for R, S, T become infinite, if the relation
rt — s2 = 0 holds at every point of the surface S. In this case the
point M describes a curve, and not a surface, for we have
II, §42] TRANSFORMATIONS 79
= ,, = 0
*,y) D*,y
and likewise
D(X, Z) = Jfo ;>* + gy - *) = _ ^ =
D(*,y) £(*, ?/)
This is precisely the case which we had not considered.
42. Ampfere's transformation. Retaining the notation of the preceding article,
let us consider the transformation
X — x, Y = q, Z — qy — z.
The relation
dZ = PdX + QdY
becomes
qdy + ydq - dz = Pdx + Qdq,
or
y dq — p dx = Pdx + Qdq.
Hence
P=-P, Q = y ;
and conversely we find
x = X, y=Q, z=QY-Z, p = - P, q = Y.
It follows that this transformation also is an involutory contact transformation.
The relation
dP = EdX+ SdY
next becomes
— r dx — s dy = R dx + S (s dx + t dy) ;
that is,
R -f Ss = - r, St = - s,
whence
Starting with the relation dQ = SdX + TdY, we find, in like manner,
r=l.
t
As an application of these formulae, let vis try to find all the functions /(x, y)
which satisfy the equation rt — s2 = 0. Let S be the surface represented by the
equation z =f(x, y), S the transformed surface, and Z = 4>(X, Y) the equation
of S. From the formulae for R it is clear that we must have
R - ^ - 0
~ S~ ~ °'
and * must be a linear function of X :
where 0 and ^ are arbitrary functions of F. It follows that
80 FUNCTIONAL RELATIONS [II, §43
and, conversely, the coordinates (a;, y, z) of a point of the surface S are given
as functions of the two variables X and Y by the formulae
x = X, y = Xt'(Y) + y(Y), z = Y[X<t>'(Y) + f'(Y)] - X<f,(Y) - t(Y).
The equation of the surface may be obtained by eliminating X and Y ; or, what
amounts to the same thing, by eliminating a between the equations
z = ay -x<t>(a)-t (a),
0 = y — x 0'(a) — ^'(a).
The first of these equations represents a moving plane which depends upon the
parameter a, while the second is found by differentiating the first with respect
to this parameter. The surfaces defined by the two equations are the so-called
developable surfaces, which we shall study later.
43. The potential equation in curvilinear coordinates. The calculation to which
a change of variable leads may be simplified in very many cases by various
devices. We shall take as an example the potential equation in orthogonal
curvilinear coordinates.* Let
F (x, y, z) - p,
FI(X, y, Z)=PI,
F*(X, y, «)=P2i
be the equations of three families of surfaces which form a triply orthogonal
system, such that any two surfaces belonging to two different families intersect
at right angles. Solving these equations for x, y, z as functions of the parame
ters p, pi, PQ, ^6 obtain equations of the form
fx-<t>(p, PI, pa),
(31) j y = <Pi(p, Pi, Pa),
l*= 02 (p, Pi, pa);
and we may take p, pt , p^ as a system of orthogonal curvilinear coordinates.
Since the three given surfaces are orthogonal, the taagents to their curves of
intersection must form a trirectangular trihedron. It follows that the equations
must be satisfied where the symbol x> indicates that we are to replace 0 by <£i,
then by 02, and add. These conditions for orthogonalism may be written in the
following form, which is equivalent to the above :
dp dpi 8p dpi dp dpi _
0,
(33)
£x dx By dy dz cz
dp dpz , n dpi Spy
— -H • • • — \j. — — -+- . .
V7
• = 0.
. Bx dx dx dx
* Lame", TraiU des coordonnees curvilignes. See also Bertrand, Traitt de Calcul
differ entiel, Vol. I, p. 181.
H, § 43] TRANSFORMATIONS 81
Let us then see what form the potential equation
ax2 dy* az2
assumes in the variables p, p1? p2. First of all, we find
dv _ dv dp 8V dpi aF apa
ax dp dx dpi dx dpz dx '
and then
a2F_ a^F/aA2 a2F ap a^. aF av
ax2 ~ a/»2 \ax/ apapi ax "aaT "a7 ax2
| 2 a2F aPl aP2 |
p^ \dx/ dpidpz ax ax api ax2
,
--
\ax/ a/>ap2 ax ax apa ax2
Adding the three analogous equations, the terms containing derivatives of the
second order like a2 V / dp dpi fall out, by reason of the relations (33), and we have
a2F . . , v a2F
(34)
A2(p) ^ + A2(P1) ~ + A2(p2) pT,
op cpi apa
where AI and A2 denote Lam&s differential parameters :
The differential parameters of the first order Ai(p), Ai(pi), Aj(p2) are easily
calculated. From the equations (31) we have
a^ ap a^ api aj^ apg _
ap ax api ax ap2 ax
a 01 ap a 01 api a 01 ap2 _
ap ax api ax apa ax
a02 a_p a02 api a02 ap2 _ . _
ap ax api ax ap2 ax
whence, multiplying by — , — -, — -, respectively, and adding, we find
dp dp dp
a0
ap _
ax ~
/a0\2 /a0A2 /a02
\dp) +\dp/ +\dpj
Then, calculating dp/dy and dp/dz in like manner, it is easy to see that
i2 /ap\2 1
«*/ \dy/ \dz/ (d<f>\ , ..„. +
82 FUNCTIONAL RELATIONS [II, §43
Let us now set
a =
dp
where the symbol $ indicates, as before, that we are to replace <f> by 0i, then
by 02) and add. Then the preceding equation and the two analogous equations
may be written
A!(P) = — , Ai(pi) = — , A!(p2) = — •
H HI HZ
Lame* obtained the expressions for A2(p), A2(pi), A2(p2) as functions of p, pi,
pa by a rather long calculation, which we may condense in the following form.
In the identity (34)
1 a2F 1 &V , 1 a2F , .8V , .8V , . , ,aF
A2 F = -—- + — —T + — -^ + A2 (p) •— + A2 (pi) — + A2(p2) — i
H dpz HI dpi H2 epjj dp dpi dpz
let us set successively V — x, V = y, V = z. This gives the three equations
i a20 i 52^ i c>20 a0
2 -- =0,
dp* HI cpj HZ cpz dp
. , • — -TT + W VF + A2 (p) iF + **(pl) -^ + ^(pz)1-— = 0,
H Cfr HI Cpj .a 2 cp2 op cpi cpz
which we need only solve for Aa(p), A2(pi), A2(p2). For instance, multiplying
by d<p/dp, d<j>i/dp, dfa/dp, respectively, and adding, we find
Moreover, we have
oe>4> 620 _ 1 ag
*-> ~d~p Up* ~ 2 lp~'
and differentiating the first of equations (32) with respect to pi, we find
~dp ~dp\ ~~ V dpi dp dpi ~ 2
In like manner we have
o£0 ev _ _ i a g2
^ ap" "ap| ~ 2 ap
and consequently
A2(p)= - -+ --
2HHi dp 2HH* dp 2H£p_ °\H1H2
H. EXS.] EXERCISES 83
Setting
*=i, zr,= », *,= £,
this formula becomes
A2(p) = ^
and in like manner we find
A2 (p!) = h\ A (log A.) , A2 (p2) = A" A (log A
api \ hhzj cpz \ fifii
Hence the formula (34) finally becomes
(35)
^ ' ^ * ll*z
ax2 a«2 az2
ra2F , a /. A \aF~l
-- h — I log - I — I
L V <>P \ *!*§/ ^/» J
*£
a
or, in condensed form,
. Fa / h dV\ , a / ftj aF\ a / &2 dF\~|
2 * 2 1 — \ / — \ »nr — / — I ^T^ s — / I '
Let us apply this formula to polar coordinates. The formulae of transforma
tion are
x = p sin0cos</>, y
where 0 and <f replace pi and p2, and the coefficients ft, fti, hq have the following
values :
& = 1 hi = -» h% = — : — -•
p p sin 0
Hence the general formula becomes
i Fa / aF\ a / . aF\ a / i aF\~|
A2F= _ - _ — Ip2sin(?— - J + — (sin^ — ) + — (- -J h
P2sine \_dp \ dp] de\ c0J a0 \sin^ a^ / J
or, expanding,
a2F i a2F i a2F 2 aF cot0aF
Ao F = -- 1 ---- h - -- h --- 1 --- '
ap2 p2 a*?2 p2sin20 a^>2 p ap p2 a<?
which is susceptible of direct verification.
EXERCISES
1. Setting u = x2 + y2 + z2, v = x + y + z, w = xy + yz + zx, the functional
determinant D (u, t>, w>) /D («, y, 2) vanishes identically. Find the relation which
exists between M, v, w.
Generalize the problem.
84 FUNCTIONAL RELATIONS pi, EXS.
2. Let
«i = —==. 1 =» •••, un -
Vi _ -r2 — a-2
1 xl — *n
Derive the equation
i, W2, • • •, M,,) . • 1
3. Using the notation
Xi = COS 0i,
X2 = sin 0i cos 02 ,
x3 = sin 0i sin 02 cos03,
Xn = sin 0i sin 02 • • • sin0n_1 cos0n,
show that
(Xl, Xg, •••, Xn) _ ^_ 1)nsinn^lSJnn-l^2Smn-2^g. . . 8in2 0n _ ! sin 0n .
4. Prove directly that the function z = F(x, y} defined by the two equations
z = ax+ yf(a) + 0(or),
0 = x + yf'(a) + 0'(a),
where a. is an auxiliary variable, satisfies the equation rt — s2 = 0, where /(a)
and 0 (a) are arbitrary functions.
5. Show in like manner that any implicit function z = F(x, y) defined by
an equation of the form
where 0 (z) and ^ (z) are arbitrary functions, satisfies the equation
rg2 - 2pqs + tp2 = 0.
6. Prove that the function z = F(x, y) defined by the two equations
z 0'(a) = [y - 0 (a)] 2, (x + a) 0'(a) = y - 0 (a),
where a is an auxiliary variable and 0 (a) an arbitrary function, satisfies the
equation pq = z.
7. Prove that the function z — F(x, y) defined by the two equations
[Z - 0 («)] 2 = Z2 (y2 _ a2), [z _ 0 ((r)] 0'(Q,) = aa.2
satisfies in like manner the equation pq = xy.
8*. Lagrange's formulae. Let y be an implicit function of the two variables
x and a, defined by the relation y = a + x<j>(y); and let u =f(y) be any func
tion of y whatever. Show that, in general,
[LAPLACE.]
II, EM.] EXERCISES 85
Note. The proof is based upon the two formulas
d F, x du~\ d F 7 \dM~l du ±i \cu
— F(u)~ = — F(u) — . — — <(,(y) — ,
da |_ dx J dx ]_ da_\ dx da
where u is any function of y whatever, and F(u) is an arbitrary function of u.
It is shown that if the formula holds for any value of n, it must hold for the
value n + I.
Setting x = 0, y reduces to a and M to /(a); and the nth derivative of u with
respect to x becomes
ff/*\
^dxn
9. If x =f(u, v), y = (J>(u, v) are two functions which satisfy the equations
dj d 0 dj d <j>
du dv dv du
show that the following equation is satisfied identically :
\ -!« i r~ ~i
M \ S" — • ,., I
:»/o da»-l\_ J
,
10. If the function F(x, y, 2) satisfies the equation
show that the function
satisfies the same equation, where A; is a constant and r2 = x2 + yz + z2.
[LORD KELVIN.]
11. If V(x, y, z) and Vi(x, y, z) are two solutions of the equation A^V = 0,
show that the function
U = F(z, y, z) + (x2 + y2 + z2) FI (x, y, z)
satisfies the equation
12. What form does the equation
(x - x8)7/"+ (1 - Sx2)?/'- xy = 0
assume when we make the transformation x = Vl — t- ?
13. What form does the equation
2
= 0
Sx2 dx dy
assume when we make the transformation x = u», y = l/v?
14*. Let 0(xi, x2, • • •, xn; MI, MS, • • •, un) be a function of the 2 n independent
variables Xi, x2, • • •, xn, MI, u2, • • •, Mn, homogeneous and of the second degree
with respect to the variables MI, M2, •••,«„. If we set
86 FUNCTIONAL RELATIONS [II, Exs.
S 0 C(t> cd>
-=pi, -=pz, •••, ~^-=pn,
CUi CUz CUn
and then take p\ , pz , • • • , pn as independent variables in the place of Ui , uz , • • • , un,
the function <f> goes over into a function of the form
d<f>
i, X2,
Derive the formulae :
15. Let N be the point of intersection of a fixed plane P with the normal MN
erected at any point M of a given surface S. Lay off on the perpendicular to the
plane P at the point N a length Nm = NM. Find the tangent plane to the
surface described by the point m, as M describes the surface S.
The preceding transformation is a contact transformation. Study the inverse
transformation.
16. Starting from each point of a given surface S, lay off on the normal to
the surface a constant length I. Find the tangent plane to the surface 2 (the
parallel surface) which is the locus of the end points.
Solve the analogous problem for a plane curve.
17*. Given a surface S and a fixed point O ; join the point 0 to any point M of
the surface S, and pass a plane OM N through OM and the normal MN to the
surface S at the point M. In this plane OMN draw through the point O a per
pendicular to the line OM, and lay off on it a length OP = OM. The point P
describes a surface 2, which is called the apsidal surface to the given surface S.
Find the tangent plane to this surface.
The transformation is a contact transformation, and the relation between the
surfaces <S and 2 is a reciprocal one. When the given surface S is an ellipsoid
and the point 0 is its center, the surface 2 is Fresnel's wave surface.
18*. Halphen's differential invariants. Show that the differential equation
dx2/ da* dx2 dx» dx* \dx*
remains unchanged when the variables x, y undergo any projective transfor
mation (§ 37).
19. If in the expression Pdx + Qdy + fidz, where P, Q, R are any functions
of x, y, z, we set
x=/(u, B, 10), y = <f> (u, v, w) , Z = ^(M, v, «>),
where «, v, w are new variables, it goes over into an expression of the form
Pidu + Q\dv + Ridw,
where PI, Qi, RI are functions of «, v, w. Show that the following equation is
satisfied identically:
gi = -P(Si ^ z)g>
I) (M, v, w)
II, Exs.] EXERCISES 87
where
/»* _ •»
\dv du
20*. Bilinear covariants. Let 0</ be a linear differential form :
where JTx, X2, • • •, -^n are functions of the n variables xlt x2, • • •, xn. Let us
consider the expression
where
and where there are two systems of differentials, d and 5. If we make any
transformation
Xi = 4>i(yi, y2, • • •, 2/n), (i = 1, 2, • • ., n),
the expression Qj goes over into an expression of the same form
Q'd= Yldyl + ••• + Yndyn,
where FI, F2, • • •, Yn are functions of yi, y2, • • •, yn. Let us also set
and
Show that H = H', identically, provided that we replace dx, and dxt, respec
tively, by the expressions
— Syi -\ 5j/2 + • • • H Syn .
The expression // is called a bilinear covariant of Qj.
21*. Beltrami's differential parameters. If in a given expression of the form
Edx* + 2Fdxdy + Gdy*,
where E, F, G are functions of the variables x and y, we make a transformation
z =/(M, v), y = <f>(u, «), we obtain an expression of the same form:
EI du2 + 2Fidudv+ Gj dv2,
88
FUNCTIONAL RELATIONS
[II, Exs.
where EI, FI, G\ are functions of u and v. Let 0(x, y) be any function of the
variables x and y, and 0i(u, v) the transformed function. Then we have, iden
tically,
ax
- 2 F + *
ex dy \dy
du dv
dv
EG- F2
El Gl - F?
IG ™ _ F —\
(E — — F —
i
8
dx
gyl |
1
dy
ax
VF.G - F2
dx \
^EG-
"F2 /
^EG - F2
^EG-
F"2
i
'-(
J1au
?i-~\
4-
1
d
^~&o~
1 du
ffj G! - Ff
- F
I
22. Schwarzian. Setting y — (ax + b) / (ex +<8), where x is a function of t and
a, 6, c, d are arbitrary constants, show that the relation
y' 2 yy
is identically satisfied, where x', x", x'", y', T/", y'" denote the derivatives with
respect to the variable t.
23*. Let u and v be any two functions of the two independent variables x and y,
and let us set
U =
au + bv + c
F =
a'w + 6'» + c'
a" « + 6" u + c" a" w + 6" u + c"
where a, &, c, • • • , c" are constants. Prove the formulae :
c*udv_G*vdu d*U dV _ cPV SU
dx2 dx gx2 dx ax2 dx dx* dx
(u, v)
d*u dv d*v du /dv
dx2 dy ex2 dy \dx dxdy dx dxdy/
(u, v)
dV a2 V d U
ax2 ~dy ~ ~d& Hy
dx dxdy ex dxdy/
(ff.F)
and the analogous formulae obtained by interchanging x and y, where
du dv du dv r dU dV dV dU
dx dy dy dx dx dy dx dy
[GOURSAT and PAINLEVE, Comptes rendus, 1887.]
CHAPTER III
TAYLOR'S SERIES ELEMENTARY APPLICATIONS
MAXIMA AND MINIMA
I. TAYLOR'S SERIES WITH A REMAINDER
TAYLOR'S SERIES
44. Taylor's series with a remainder. In elementary texts on the
Calculus it is shown that, if f(x) is an integral polynomial of
degree n, the following formula holds for all values of a and h :
This development stops of itself, since all the derivatives past the
(n + l)th vanish. If we try to apply this formula to a function
/(x) which is not a polynomial, the second member contains an
infinite number of terms. In order to find the proper value to
assign to this development, we will first try to find an expression
for the difference
f> li 2 J)n
f(a + h) -f(a) - 2 f(a) - f^ /"(a) ___/W(a),
with the hypotheses that the function /(#), together with its first n
derivatives /'(a:), f"(x), • • • , f^n)(x), is continuous when x lies in the
interval (a, a -f A), and that f(n\x) itself possesses a derivative
/(" + J)(x) in the same interval. The numbers a and a -f h being
given, let us set
(2)
where p is any positive integer, and where P is a number which is
defined by this equation itself. Let us then consider the auxiliary
function
89
90 TAYLOR'S SERIES [in, §44
=f(a + A) -/(») - =f,(x) _ ~
_ O + h ~ *)" /-A _ (a + h-x
"
J
1.2- "ii 1.2..-
It is clear from equation (2), which defines the number P, that
and it results from the hypotheses regarding f(x) that the func
tion <£(x) possesses a derivative throughout the interval (a, a -f A).
Hence, by Rolle's theorem, the equation </>'(#) = 0 must have a root
a + Oh which lies in that interval, where 0 is a positive number
which lies between zero and unity. The value of <t>'(x), after some
easy reductions, turns out to be
The first factor (a -f h — x~)p~l cannot vanish for any value of x
other than a + h. Hence we must have
P= hn-p + l(l - 0)"-* + !/(» + J> (a + Ofy, where 0<^<1;
whence, substituting this value for P in equation (2), we find
(3) /I
where
JL =
0
. 2 • • • n .p
We shall call this formula Taylor's scries with a remainder, and
the last term or Rn the remainder. This remainder depends upon the
positive integer p, which we have left undetermined. In practice,
about the only values which are ever given to p are p = n + 1 and
p = 1. Setting p = n + 1, we find the following expression for the
remainder, which is due to Lagrange :
setting p — 1, we find
Ill, §44] TAYLOR'S SERIES WITH A REMAINDER 91
an expression for the remainder which is due to Cauchy. It is
clear, moreover, that the number 0 will not be the same, in general,
in these two special formulae. If we assume further that /(n + 1)(a:)
is continuous when x = a, the remainder may be written in the form
where e approaches zero with h.
Let us consider, for definiteness, Lagrange's form. If, in the gen
eral formula (3), n be taken equal to 2, 3, 4, • • • , successively, we
get a succession of distinct formulae which give closer and closer
approximations for f(a -f- A) for small values of h. Thus for n = 1
we find
1 1.2
which shows that the difference
/(* + *) -/(a) -*/*(«)
is an infinitesimal of at least the second order with respect to h,
provided that /" is finite near x = a. Likewise, the difference
_/•/ , 7 \ _/»/ \ '*" _/»l / \ *^ /»/// \
f(a + A) — f(a) — - f '(a) — - — - f (a)
«/\ / «/ \ / 1 V \ / 1 O */ \ /
J. JL » *«
is an infinitesimal of the third order ; and, in general, the expression
A a + h) — f(a) — - f'(a) — • • • : f(n)(a)
J v \ / ~1 v \ / -w! \ f
is an infinitesimal of order n + 1. But, in order to have an exact
idea of the approximation obtained by neglecting R, we need to
know an upper limit of this remainder. Let us denote by M* an
upper limit of the absolute value of y(" + 1)(#) in the neighborhood
of x = a, say in the interval (a — 17, a -f rj). Then we evidently have
Kti< i^r1 M)
provided that | h | < 77.
* That is, 3/>|/(« + J)(z) I when |z — a\<ij. The expression " the upper limit,"
defined in § 68, must be carefully distinguished from the expression " an upper limit,"
which is used here to denote a number greater than or equal to the absolute value of
the function at any point in a certain interval. In this paragraph and in the next
/(» + i)(a;) is supposed to have an upper limit near x — a. — TRANS.
92 TAYLOR'S SERIES [III, §45
45. Application to curves. This result may be interpreted geomet
rically. Suppose that we wished to study a curve C, whose equa
tion is y =f(x), in the neighborhood of a point A, whose abscissa
is a. Let us consider at the same time an auxiliary curve C", whose
equation is
A line x = a -f- h, parallel to the axis of y, meets these two curves
in two points M and M', which are near A. The difference of their
ordinates, by the general formula, is equal to
This difference is an infinitesimal of order not less than n -+- 1 ; and
consequently, restricting ourselves to a small interval (a — 77, a + rj),
the curve C sensibly coincides with the curve C'. By taking larger
and larger values of n we may obtain in this way curves which
differ less and less from the given curve C; and this gives us a
more and more exact idea of the appearance of the curve near the
point A.
Let us first set n = \. Then the curve C" is the tangent to the
curve C at the point A :
and the difference between the ordinates of the points M and M '
of the curve and its tangent, respectively, which have the same
abscissa a -f h, is
Let us suppose that /"(«) =£ 0, which is the case in general. The
preceding formula may be written in the form
where c approaches zero with h. Since f"(a) =£ 0, a positive num
ber rj can be found such that e | < | /"(a) | , when h lies between — rj
and + 77. For such values of h the quantity /"(«) + e will have
the same sign as /"(a)> an(i hence y — Y will also have the same
sign as /"(a). If /"(a) is positive, the ordinate y of the curve is
Ill, § 46] TAYLOR'S SERIES WITH A REMAINDER 93
greater than the ordinate F of the tangent, whatever the sign of h ;
and the curve C lies wholly above the tangent, near the point A.
On the other hand, if /"(a) is negative, y is less than Y, and the
curve lies entirely below the tangent, near the point of tangency.
If f"(a) = 0, let /(p)(«) be the first succeeding derivative which
does not vanish for x = a. Then we have, as before, if f(p)(x) is
continuous when x = a,
and it can be shown, as above, that in a sufficiently small interval
(a — ^ a -f- 17) the difference y — Y has the same sign as the product
Ap/(p)(a). When p is even, this difference does not change sign
with h, and the curve lies entirely on the same side of the tangent,
near the point of tangency. But if p be odd, the difference y — Y
changes sign with h, and the curve C crosses its tangent at the
point of tangency. In the latter case the point A is called a point
of inflection ; it occurs, for example, if f'"(a) =£ 0.
Let us now take n = 2. The curve C' is in this case a parabola :
Y =f(a) + (x
whose axis is parallel to the axis of y\ and the difference of the
ordinates is
If /'"(a) does not vanish, y — Y has the same sign as A8/'"(«) for
sufficiently small values of h, and the curve C crosses the parabola
C' at the point A. This parabola is called the osculatory parabola
to the curve C ; for, of the parabolas of the family
Y = mxz + nx + p,
this one comes nearest to coincidence with the curve C near the
point A (see § 213).
46. General method of development. The formula (3) affords a
method for the development of the infinitesimal f(a + K) ~f(a}
according to ascending powers of h. But, still more generally, let
x be a principal infinitesimal, which, to avoid any ambiguity, we
94 TAYLOR'S SERIES [III, §46
will suppose positive ; and let y be another infinitesimal of the
form
(4) y = AlX»i + A2x»* + ...+x»P (Ap + c),
where nl} nz, •• • , np are ascending positive numbers, not necessarily
integers, Al} At, ••-, Ap are constants different from zero, andc is
another infinitesimal. The numbers HI, Al, n2, A2, ••• may be cal
culated successively by the following process. First of all, it is
clear that HI is equal to the order of the infinitesimal y with
respect to x, and that Av is equal to the limit of the ratio y/xni when
x approaches zero. Next we have
y — A:xn^ = ui = Azx"* -| ---- + (Ap + c) x"p,
which shows that nz is equal to the order of the infinitesimal MI?
and A2 to the limit of the ratio u^/xn-i. A continuation of this
process gives the succeeding terms. It is then clear that an infini
tesimal y does not admit of two essentially different developments of
the form (4). If the developments have the same number of terms,
they coincide ; while if one of them has p terms and the other
p + q terms, the terms of the first occur also in the second. This
method applies, in particular, to the development of f(a + A) —f(a)
according to powers of h ; and it is not necessary to have obtained
the general expression for the successive derivatives of the func
tion f(x) in advance. On the contrary, this method furnishes
us a practical means of calculating the values of the derivatives
Examples. Let us consider the equation
(5) F(x, y) = Axn + By + xy<S>(x, y) + Cxn + l + • • • + Dy2 + - - - = 0,
where 4> (x, y) is an integral polynomial in x and y, and where the
terms not written down consist of two polynomials P(x) and Q(y),
which are divisible, respectively, by xn + 1 and y2. The coefficients A
and B are each supposed to be different from zero. As x approaches
zero there is one and only one root of the equation (5) which ap
proaches zero (§ 20). In order to apply Taylor's series with a
remainder to this root, we should have to know the successive deriv
atives, which could be calculated by means of the general rules.
But we may proceed more directly by employing the preceding
method. For this purpose we first observe that the principal part
Ill, §46] TAYLOR'S SERIES WITH A REMAINDER 95
of the infinitesimal root is equal to — (.4 /E)xn. For if in the equa
tion (5) we make the substitution
y = •
and then divide by xn, we obtain an equation of the same form :
(cc, yi)
which has only one term in ylt namely By^. As x approaches zero
the equation (6) possesses an infinitesimal root in ylt and conse
quently the infinitesimal root of the equation (5) has the principal
part — (A/B)xn, as stated above. Likewise, the principal part of
?/! is — (A±l B)x**] and we may set
where yz is another infinitesimal whose principal part may be found
by making the substitution
in the equation (6).
Continuing in this way, we may obtain for this root y an expres
sion of the form
(ap + e)x
.n + MJ H 1- n
n
which we may carry out as far as we wish. All the numbers
H!, nz, •••, np are indeed positive integers, as they should be, since
we are working under conditions where the general formula (3) is
applicable. In fact the development thus obtained is precisely the
same as that which we should find by applying Taylor's series with
a remainder, where a = 0 and h = x.
Let us consider a second example where the exponents are not
necessarily positive integers. Let us set
y
96 TAYLOR'S SERIES
[III, §46
where a, ft, y, • • • and fa, yu • • • are two ascending series of positive
numbers, and the coefficient A is not zero. It is clear that the prin
cipal part of y is Axa, and that we have
tt _ + Cx* + • • • - A xa(Blx^ + Cix* + •••)
y — J]_ 00 — ' — - • — '- • ••— •— y
1 + B^ + dx* H ----
which is an expression of the same form as the original, and whose
principal part is simply the term of least degree in the numerator.
It is evident that we might go on to find by the same process as
many terms of the development as we wished.
Let / (x) be a function which possesses n + 1 successive derivatives. Then
replacing a by x in the formula (3), we find
f(x + h) -/(x) + -f(x) + ~f"(x) + ..-+ *" [/00(x) + e] ,
l . fi 1 . 2 • • • n
where e approaches zero with h. Let us suppose, on the other hand, that we
had obtained by any process whatever another expression of the same form for
f(x + h) = /(x) + hfa (x) + A202 (X) + . . . + hn faty + e/] .
These two developments must coincide term by term, and hence the coefficients
0i> 02> • • •, <t>n are equal, save for certain numerical factors, to the successive
derivatives of /(x) :
, -_.
1.2 1. 2- • -n
This remark is sometimes useful in the calculation of the derivatives of certain
functions. Suppose, for instance, that we wished to calculate the nth derivative
of a function of a function :
y=f(u), where u = 0(x).
Neglecting the terms of order higher than n with respect to h, we have
* = 0(X + h) - 0(X) = J 0'(X) + -^ 0"(X) + • • • + , ** 0W(Z);
1 . 2i 1 . 2 • • • n
and likewise neglecting terms of order higher than n with respect to k,
"
f(u + k) ~f(u) = f(u)
i . * 1 . 2 • • • n
If in the right-hand side k be replaced by the expression
^ + *"(x) + • • • + '
T
i . & • • • n
and the resulting expression arranged according to ascending powers of A, it is
evident that the terms omitted will not affect the terms in h, h2, • • • , hn. The
HI, §47] TAYLOR'S SERIES WITH A REMAINDER 97
coefficient of /i", for instance, will be equal to the nth derivative of /[</>(£)]
divided by 1 . 2 • • • n ; and hence we may write
where Ai denotes the coefficient of hn in the development of
For greater detail concerning this method, the reader is referred to Hermite's
Cours d' Analyse (p. 59).
47. Indeterminate forms.* Let f(x) and </> (x) be two functions
which vanish for the same value of the variable x = a. Let us try
to find the limit approached by the ratio
f(a + K)
<f>(a + h)
as h approaches zero. This is merely a special case of the problem
of finding the limit approached by the ratio of two infinitesimals
The limit in question may be determined immediately if the prin
cipal part of each of the infinitesimals is known, which is the case
whenever the formula (3) is applicable to each of the functions
/(cc) and <f> (x) in the neighborhood of the point a. Let us suppose
that the first derivative of f(x) which does not vanish for x = a is
that of order p, f(p\a) ; and that likewise the first derivative of
<£ (cc) which does not vanish for x = a is that of order q, <£(s)(a).
Applying the formula (3) to each of the functions f(x) and <£ (x)
and dividing, we find
where c and e' are two infinitesimals. It is clear from this result
that the given ratio increases indefinitely when h approaches zero, if
q is greater than p ; and that it approaches zero if q is less than p.
If q = p, however, the given ratio approaches /(p)(a)/^(7)(a) as ^s
limit, and this limit is different from zero.
Indeterminate forms of this sort are sometimes encountered in finding the
tangent to a curve. Let
x =/(<), y = *(t), z = *(l)
* See also § 7.
98 TAYLOR'S SERIES [m, J48
be the equations of a curve C in terms of a parameter t. The equations of the
tangent to this curve at a point M, which corresponds to a value t0 of the param
eter, are, as we saw in § 5,
Z -
f'(tv) *'(*o) f'(«o)
These equations reduce to identities if the three derivatives /'(£), <f>'(t), $'(t) all
vanish for t = t0. In order to avoid this difficulty, let us review the reasoning
by which we found the equations of the tangent. Let M' be a point of the
curve C near to M, and let to + h be the corresponding value of the parameter.
Then the equations of the secant MM' are
For the sake of generality let us suppose that all the derivatives of order less
than p (p> 1) of the functions /(£), <t> (t), \f/ (t) vanish for t = t0, but that at least
one of the derivatives of order p, say /<*>> (£0), is not zero. Dividing each of the
denominators in the preceding equations by hp and applying the general for
mula (3), we may then write these equations in the form
to) + e &>J) (to) + e' f("> (to) + e"
where e, e', e" are three infinitesimals. If we now let h approach zero, these
equations become in the limit
in which form all indetermination has disappeared.
The points of a curve C where this happens are, in general, singular points
where the curve has some peculiarity of form. Thus the plane curve whose
equations are
X = «», y = <3
passes through the origin, and dx/dt = dy / dt = 0 at that point. The tangent
is the axis of x, and the origin is a cusp of the first kind.
48. Taylor's series. If the sequence of derivatives of the function
f(x) is unlimited in the interval (a, a -f h), the number n in the
formula (3) may be taken as large as we please. If the remainder
Rn approaches zero when n increases indefinitely, we are led to write
down the following formula :
which expresses that the series
/(«) + \ /'(«) + • • • + iT^r^ ^"W + ' '
Ill, §48] TAYLOR'S SERIES WITH A REMAINDER 99
is convergent, and that its " sum " * is the quantity f(a + h). This
formula (7) is Taylor's series, properly speaking. But it is not justi
fiable unless we can show that the remainder Rn approaches zero when
n is infinite, whereas the general formula (3) assumes only the exist
ence of the first n + 1 derivatives. Replacing a by x, the equation
(7) may be written in the form
Or, again, replacing h by x and setting a, = 0, we find the formula
(8) /(*) =/(0) + /'(O) + • - • + -- /
This latter form is often called Maclaurin 's series; but it should
be noticed that all these different forms are essentially equivalent.
The equation (8) gives the development of a function of x accord
ing to powers of x ; the formula (7) gives the development of a func
tion of h according to powers of h : a simple change of notation is
all that is necessary in order to pass from one to the other of these
forms.
It is only in rather specialized cases that we are able to show
that the remainder Rn approaches zero when n increases indefinitely.
If, for instance, the absolute value of any derivative whatever is less
than a fixed number M when x lies between a and a + h, it follows,
from Lagrange's form for the remainder, that
I h\" + 1
I*' ' '^l. 2- ..(» + !)'
an inequality whose right-hand member is the general term of a
convergent series. f Such is the case, for instance, for the functions
ex, sin x, cos x. All the derivatives of ex are themselves equal to
ex, and have, therefore, the same maximum in the interval con
sidered. In the case of sin x and cos x the absolute values never
exceed unity. Hence the formula (7) is applicable to these three
functions for all values of a and h. Let us restrict ourselves to
the form (8) and apply it first to the function f(x) = ex. We find
* That is to say, the limit of the sum of the first n terms as n becomes infinite.
For a definition of the meaning of the technical phrase " the sum of a series," see
§ 157.— TRANS.
t The order of choice is a, h, M, n, not a, h, n, M. This is essential to the con
vergence of the series in question. — TRANS.
100 TAYLOR'S SERIES [III, §49
and consequently we have the formula
which applies to all values, positive or negative, of x. If a is any
positive number, we have ax = erl°Ka, and the preceding formula
becomes
Let us now take f(x) = sin x. The successive derivatives form a
recurrent sequence of four terms cos x, — sin x, — cos a;, sin x ; and
their values for x = 0 form another recurrent sequence 1, 0, — 1, 0.
Hence for any positive or negative value of x we have
(11)
and, similarly,
(12) .o.,-l- + J!L_
Let us return to the general case. The discussion of the remain
der Rn is seldom so easy as in the preceding examples; but the
problem is somewhat simplified by the remark that if the remain
der approaches zero the series
necessarily converges. In general it is better, before examining
Rn, to see whether this series converges. If for the given values of
a and h the series diverges, it is useless to carry the discussion
further ; we can say at once that Rn does not approach zero when n
increases indefinitely.
49. Development of log(l 4- x). The function log(l + x), together
with all its derivatives, is continuous provided that x is greater
than — 1. The successive derivatives are as follows :
HI, §49] TAYLOR'S SERIES WITH A REMAINDER 101
_ i
Let us see for what values of a? Maclaurin's formula (8) may be
applied to this fu action. Writing first the series with a remainder,
we have, under any circumstances,
*) = - + + ••• + (- !)-'• + *,
The remainder Rn does not approach zero unless the series
converges, which it does only for the values of x between — 1 and
+ 1, including the upper limit -f 1. When x lies in this interval
the remainder may be written in the Cauchy form as follows :
_ iyl.2-.-n
~
1.2
or
(
1 — (9
Let us consider first the case where | x < 1. The first factor x
approaches zero with x, and the second factor (1 — 6)/(l + Ox) is
less than unity, whether x be positive or negative, for the numer
ator is always less than the denominator. The last factor remains
finite, for it is always less than 1/(1 — |x|). Hence the remainder
Rn actually approaches zero when n increases indefinitely. This
form of the remainder gives us no information as to what happens
when x = 1 ; but if we write the remainder in Lagrange's form,
it is evident that Rn approaches zero when n increases indefinitely.
An examination of the remainder for x = — 1 would be useless,
1:02 TAYLOR'S SERIES [III, §49
sinf.'e the series diverges for that value of x. We have then, when
x lies between — 1 and -f 1, the formula
(13) log(l+*)^-f + |3- . + (-l)-i£ + ....
This formula still holds when x = 1, which gives the curious
relation
(14) Iog2«l-|+|-j+. . + (-1)-^ + ....
The formula (13), not holding except when x is less than or equal
to unity, cannot be used for the calculation of logarithms of whole
numbers. Let us replace x by — x. The new formula obtained,
still holds for values of x between — 1 and + 1 ; and, subtracting
the corresponding sides, we find the formula
«2 + + f--- + T-T
1 — x] \1 3 o 2 n + 1
When x varies from 0 to 1 the rational fraction (1 +#)/(! — a:)
steadily increases from 1 to + <x>, and hence we may now easily cal
culate the logarithms of all integers. A still more rapidly con
verging series may be obtained, however, by forming the difference
of the logarithms of two consecutive integers. For this purpose
let us set
1+ x N + 1 1
or x =
1-x N 2N+1
Then the preceding formula becomes
an equation whose right-hand member is a series which converges
very rapidly, especially for large values of N.
Note. Let us apply the general formula (3) to the function log (1 + x), setting
a = 0, h = x, n = 1, and taking Lagrange's form for the remainder. We find in
this way
x2
log(l + x) = x ---- •
Ill, §49] TAYLOR'S SERIES WITH A REMAINDER 103
If we now replace x by the reciprocal of an integer n, this may be written
n 2n2
where 0n is a positive number less than unity. Some interesting consequences
may be deduced from this equation.
1) The harmonic series being divergent, the sum ' v-r - \ \\/ % .
31 r \
1 1 1
+n
, \
increases indefinitely with n. But the difference
'vw
2n — log n
approaches a finite limit. For, let us write this difference in the form
I n + 1\ n+ 1
-log ) + log
n n / n
Now 1 / p — log (1 + 1 / ' p) is the general term of a convergent series, for by the
equation above
which shows that this term is smaller than the general term of the convergent
series 2(1 /p2). When n increases indefinitely the expression
n+1 /n 1\
log = log ( 1 + -
n V */
approaches zero. Hence the difference under consideration approaches a finite
limit, which is called Euler's constant. Its exact value, to twenty places of
decimals, is C = 0.57721566490153286060.
2) Consider the expression
n+1 n + 2 n + p
where n and p are two positive integers which are to increase indefinitely. Then
7?e may write
2 n + p/
1 1
2 + • • • + — — = log (n
1 1
2 + ' n
104 TAYLOR'S SERIES [III, §50
where pn+P and pn approach the same value C when n and p increase indefi
nitely. Hence we have also
Now the difference pn+p — pn approaches zero. Hence the sum 2 approaches
no limit unless the ratio p/n approaches a limit. If this ratio does approach a
limit a, the sum S approaches the limit log (1 + a).
Setting p — n, for instance, we see that the sum
n+ 1 n+2 2n
approaches the limit log 2.
50. Development of (1 + x)m. The function (1 + x)m is denned and
continuous, and its derivatives all exist and are continuous func
tions of x, when 1 + x is positive, for any value of m ; for the
derivatives are of the same form as the given function :
fM(x) = m(m - 1) - - - (m - n + 1) (1 +
/(» + 1)(x) = m(m - 1) • • • (m - n) (1 + a)"
Applying the general formula (3), we find
-...-
1.2- ••» "'
and, in order that the remainder Rn should approach zero, it is first
of all necessary that the series whose general term is
(m — 1) • • • (m — n + 1)
1.2-.-W
should converge. But the ratio of any term to the preceding is
m — n -\- 1
x
which approaches — x as n increases indefinitely. Hence, exclud
ing the case where m is a positive integer, which leads to the ele
mentary binomial theorem, the series in question cannot converge
unless \x < 1. Let us restrict ourselves to the case in which I a; I < 1.
Ill, §50] TAYLOR'S SERIES WITH A REMAINDER 105
To show that the remainder approaches zero, let us write it in the
Cauchy form :
• 1.2...
The first factor
m(m — 1) • •• (m — n~) n + l
1.2..-W
approaches zero since it is the general term of a convergent
series. The second factor (1 — #)/(! -f &c) is less than unity; and,
finally, the last factor (1 + Ox)m~l is less than a fixed limit. For,
if m - 1 > 0, we have (1 + ftc)"—1 < 2'"-1 ; while if m - 1< 0,
(1 + Ox)m~l< (1 — ici)"1"1. Hence for every value of x between
— 1 and + 1 we have the development
-.-
We shall postpone the discussion of the case where x = ± 1.
In the same way we might establish the following formulae
, 1 xa , 1 . 3 x6 ,
arcsm* = * + -- + ^- + ...
.5...(2r? -1) x2" + 1
2.4.6--.2w 2w-|-l "'
X8 iC5 iC7 CC2fl + 1
- + ¥-y + ... + (-l)»^TI+---,
which we shall prove later by a simpler process, and which hold
for all values of x between — 1 and + 1.
Aside from these examples and a few others, the discussion of
the remainder presents great difficulty on account of the increas
ing complication of the successive derivatives. It would therefore
seem from this first examination as if the application of Taylor's
series for the development of a function in an infinite series were of
limited usefulness. Such an impression would, however, be utterly
false ; for these developments, quite to the contrary, play a funda
mental role in modern Mathematical Analysis. In order to appre
ciate their importance it is necessary to take another point of
view and to study the properties of power series for their own
106 TAYLOR'S SERIES [III, §'5J
sake, irrespective cf their origin. We shall do this in several of
the following chapters.
Just now we will merely remark that the series
may very well be convergent without representing the function
f(x) from which it was derived. The following example is due to
Cauchy. Let /(*) = e~1'*. Then /'(*) = (2/x*)e-l"> ; and, in
general, the nth derivative is of the form
where P is a polynomial. All these derivatives vanish for x = 0,
for the quotient of e~l/3* by any positive power of x approaches
zero with x.* Indeed, setting x = 1/z, we may write
and it is well known that ez*/zm increases indefinitely with z, no
matter how large m may be. Again, let <f> (x) be a function to which
the formula (8) applies :
Setting F(x) = <fr(x') + e~llx\ we find
F(0) . <*> (0), F'(0) - *'(0), • • • , F<">(0) - ^->(0), • • • ,
and hence the development of F(x) by Maclaurin's series would
coincide with the preceding. The sum of the series thus obtained
represents an entirely different function from that from which the
series was obtained.
In general, if two distinct functions f(x) and <f> (a;), together with
all their derivatives, are equal for x = 0, it is evident that the
*It is tacitly assumed that /(O) = 0, which is the only assignment which would
render/(:c) continuous at x = 0. But it should be noticed that no further assignment
is necessary for /'(a:), etc., at x = 0. For
,,,m lim /(a) -/(O) _ ft
= x = 0 ^~~ - °'
which defines /'(x) at x — 0 and makes /'(z) continuous at a: = 0, etc. — TRANS.
Ill, §61] TAYLOR'S SERIES WITH A REMAINDER 107
Maclaurin series developments for the two functions cannot both
be valid, for the coefficients of the two developments coincide.
51. Extension to functions of several variables. Let us consider, for
definiteness, a function o> = f(x, y, z) of the three independent vari
ables x, y, z, and let us try to develop f(x -f h, y -f k, z + I) accord
ing to powers of h, k, I, grouping together the terms of the same
degree. Cauchy reduced this problem to the preceding by the fol
lowing device. Let us give x, y, z, h, k, I definite values and let
us set
<f> (£) =f(x + ht,y + kt, z + ft),
where t is an auxiliary variable. The function <£(£) depends on t
alone ; if we apply to it Taylor's series with a remainder, we find
(17)
f— *<">(0) + ? r^ -T; *<-+'>(*),
where <£(0), <£'(0), •••, <£(n)(0) are the values of the function <f>(f)
and its derivatives, for t — 0; and where <£(" + 1)(0£) is the value of
the derivative of order n + 1 for the value &t, where 0 lies between
zero and one. But we may consider <£ (f) as a composite function of
t, $(£) =/(w, v, w), the auxiliary functions
u = x 4- ht, v = y + kt, w = K -f It
being linear functions of t. According to a previous remark, the
expression for the differential of order m, dm<f>, is the same as if t/,
v, w were the independent variables. Hence we have the symbolic
equation
which may be written, after dividing by dtm, in the form
A + |£t + LC
CV CW
For t = 0, u, v, w reduce, respectively, to x, y, z, and the above
equation in the same symbolism becomes
108 TAYLOR'S SERIES [III, §52
Similarly,
(n + l)
where cc, y, z are to be replaced, after the expression is developed, by
x + Oht, y + 6kt, z + Bit,
respectively. If we now set t = 1 in (17), it becomes
,dx
(18)1
1f\ Id '-!/-> - I *-;
. 2 • • • n \cx cy cz
The remainder Rn may be written in the form
n+1)
where x, y, z are to be replaced by x + 6h, y + 6k, z + 01 after the
expression is expanded.*
This formula (18) is exactly analogous to the general formula
(3). If for a. given set of values of cc, y, z, h, k, I the remainder Rn
approaches zero when n increases indefinitely, we have a develop
ment of f(x + h, y + k, z + I) in a series each of whose terms is a
homogeneous polynomial in h, k, I. But it is very difficult, in gen
eral, to see from the expression for Rn whether or not this remainder
approaches zero.
52. From the formula (18) it is easy to draw certain conclusions
analogous to those obtained from the general formula (3) in the
case of a single independent variable. For instance, let z =f(x, y)
be the equation of a surface S. If the function f(x, y), together
with all its partial derivatives up to a certain order n, is continuous
in the neighborhood of a point (XQ, y0), the formula (18) gives
• / df ,
/(«• + h, y0 + k) = f(x0, y0~) + I h ^- -f fc 7
1 . 2
Restricting ourselves, in the second member, to the first two terms,
then to the first three, etc., we obtain the equation of a plane, then
* It is assumed here that all the derivatives used exist and are continuous. — TRANS.
Ill, §52] TAYLOR'S SERIES WITH A REMAINDER 109
that of a paraboloid, etc., which differ very little from the given sur
face near the point (x0) y0)- The plane in question is precisely the
tangent plane ; and the paraboloid is that one of the family
* = Ax2 + 2 Bxy + Cy*
which most nearly coincides with the given surface S.
The formula (18) is also used to determine the limiting value of
a function which is given in indeterminate form. Let f(x, y) and
<£ (#, T/) be two functions which both vanish for x = a, y = b, but
which, together with their partial derivatives up to a certain order,
are continuous near the point (a, £). Let us try to find the limit
approached by the ratio
when x and y approach a and b, respectively. Supposing, first, that
the four first derivatives df/da, df/8b, d<f>/8a, 8<f>/db do not all
vanish simultaneously, we may write
k[T*-+<} + k(%-
K) _
<(>(a + h,b + k) i /d<f> . \, ,
h\Ta+^) + k\db
where e, c', c,, e{ approach zero with h and &. When the point
(x, y) approaches (a, b~), h and k approach zero ; and we will sup
pose that the ratio k/h approaches a certain limit a, i.e. that the
point (x, y) describes a curve which has a tangent at the point (a, b~).
Dividing each of the terms of the preceding ratio by A, it appears
that the fraction f(x, y)/<$>(x, y) approaches the limit
o^~ "+" a ~oT
ca do
-Z -- P OL -^r-
ca cb
This limit depends, in general, upon a, i.e. upon the manner in
which x and y approach their limits a and b, respectively. In order
that this limit should be independent of a it is necessary that the
relation
_
da db db da
should hold ; and such is not the case in general.
110 TAYLOR'S SERIES [HI, §53
If the four first derivatives df/Sa, df/Sb, d<l>/da, 8<f>/db vanish
simultaneously, we should take the terms of the second order in the
formula (18) and write
*
f(a + h, b + K) _
where e, e', c", cu e/, e'/ are infinitesimals. Then, if a be given the
same meaning as above, the limit of the left-hand side is seen to be
V? + 2 f-fj a H- %-? a2
£a <ya cb co
which depends, in general, upon a.
II. SINGULAR POINTS MAXIMA AND MINIMA
53. Singular points. Let (x0, ?/0) be the coordinates of a point M0
of a curve C whose equation is F(x, y~) = 0. If the two first par
tial derivatives 8F/dx, 8F/dy do not vanish simultaneously at this
point, we have seen (§ 22) that a single branch of the curve C passes
through the point, and that the equation of the tangent at that
point is
where the symbol dp + qF /dx$ dyl denotes the value of the derivative
fip + vp /8xp di/* for x = x0, y = y0. If dF/dxQ and dF /dy0 both van
ish, the point (x0, y0) is, in general, a singular point* Let us suppose
that the three second derivatives do not all vanish simultaneously
for x = x0, y — y0, and that these derivatives, together with the third
derivatives, are continuous near that point. Then the equation of
the curve may be written in the form.
* That is, the appearance of the curve is, in general, peculiar at that point. For an
exact analytic definition of a singular point, see § 192. — TRANS.
Ill, §53] SINGULAR POINTS MAXIMA AND MINIMA ' 111
(19)
;_ g \24-2 ^
1 [OF .
I / 'y*
cF
where x and y are to be replaced in the third derivatives by
X() + Q(x- B0) and ?/0 + 0(y - y0), respectively. We may assume
that the derivative d*F/dy$ does not vanish; for, at any rate, we
could always bring this about by a change of axes. Then, setting
y — y0 = t (x — ar0) and dividing by (x — ic0)2, the equation (19)
becomes
(20)
1
0 fyo
- 0,
where P (x — x0, t*) is a function which remains finite when x
approaches x0. Now let ti and #2 be the two roots of the equation
If these roots are real and unequal, i.e. if
2
the equation (20) may be written in the form
(t - £,) (t - *2) + (x - x,) P = 0.
For x = x0 the above quadratic has two distinct roots t = tl9 t = £2.
As x approaches x0 that equation has two roots which approach #t
and £2, respectively. The proof of this is merely a repetition of
the argument for the existence of implicit functions. Let us set
t = tl-\- u, for example, and write down the equation connecting x
and u:
where Q (x, u) remains finite, while x approaches x0 and u approaches
zero. Let us suppose, for definiteness, that tl — tz > 0 ; and let M
denote an upper limit of the absolute value of Q(x, «), and ra a
lower limit of tt — t2 + u, when x lies between x0 — h and x0 + h,
112 TAYLOR'S SERIES [III, §53
and u between — h and -f h, where h is a positive number less than
ti — t2. Now let c be a positive number less than h, and rj another
positive number which satisfies the two inequalities
m
77 < h, rj < — e.
If a; be given such a value that x — x0 \ is less than 77, the left-hand
side of the above equation will have different signs if — e and then
-+- c be substituted for u. Hence that equation has a root which
approaches zero as x approaches x0, and the equation (19) has a
root of the form
V = !/o + (a - *o) (*i + a),
where a approaches zero with x — x0. It follows that there is one
branch of the curve C which is tangent to the straight line
y - y<> = *i (* - z<>)
at the point (x0, T/O).
In like manner it is easy to see that another branch of the
curve passes through this same point tangent to the straight line
y — yQ = t2(x — x0). The point M0 is called a double point; and
the equation of the system of tangents at this point may be found
by setting the terms of the second degree in (x — x0), (y — y0) in
(19) equal to zero.
If
the point (cc0, T/O) is called an isolated double point. Inside a suffi
ciently small circle about the point Af0 as center the first member
F(x, ?/) of the equation (19) does not vanish except at the point M0
itself. For, let us take
x = x0 + p cos <f>, y = ?/0 + p sin <f>
as the coordinates of a point near M0. Then we find
,2
" - ~ cos <^> sin 4» + - sin2^ + PL
where L remains finite when p approaches zero. Let H be an upper
limit of the absolute value of L when p is less than a certain posi
tive number r. For all values of <£ between 0 and 2?r the expression
c2F
cos «A sin d> + -T-T- sin2
Ill, §53] SINGULAR POINTS MAXIMA AND MINIMA 113
has the same sign, since its roots are imaginary. Let m be a lower
limit of its absolute value. Then it is clear that the coefficient
of p2 cannot vanish for any point inside a circle of radius p<m/H.
Hence the equation F(x, y) = 0 has no root other than p = 0, i.e.
x = x0, y = y0, inside this circle.
In case we have
dx08y0
the two tangents at the double point coincide, and there are, in gen
eral, two branches of the given curve tangent to the same line, thus
forming a cusp. The exhaustive study of this case is somewhat
intricate and will be left until later. Just now we will merely
remark that the variety of cases which may arise is much greater
than in the two cases which we have just discussed, as will be seen
from the following examples.
The curve y2 = xs has a cusp of the first kind at the origin, both
branches of the curve being tangent to the axis of x and lying on
different sides of this tangent, to the right of the y axis. The
curve y2 — 2x2y + x* — x5 = 0 has a cusp of the second kind, both
branches of the curve being tangent to the axis of x and lying on
the same side of this tangent ; for the equation may be written
y = x2 ± z%
and the two values of y have the same sign when x is very small,
but are not real unless x is positive. The curve
has two branches tangent to the x axis at the origin, which do not
possess any other peculiarity ; for, solving for y, the equation becomes
3 x2 ± x2 V8 - x2
y~ 1+x2
and neither of the two branches corresponding to the two signs
before the radical has any singularity whatever at the origin.
It may also happen that a curve is composed of two coincident
branches. Such is the case for the curve represented by the
equation
When the point (x, y) passes across the curve the first member F(x, y)
vanishes without changing sign.
11-4 TAYLOR'S SERIES [III, §54
Finally, the point (cc0, y0) may be an isolated double point. Such
is the case for the curve y2 + x4 + y4 = 0, on which the origin is an
isolated double point.
54. In like manner a point M0 of a surface S, whose equation is
F(x> y> «) = 0, is, in general, a singular point of that surface if the
three first partial derivatives vanish for the coordinates x0, y0, z0 of
that point :
dF _ ZF _ CF
•5 — — w, - — — 0, 7 — = 0.
CXQ CIJ^ CZQ
The equation of the tangent plane found above (§ 22) then reduces
to an identity ; and if the six second partial derivatives do not all
vanish at the same point, the locus of the tangents to all curves on
the surface S through the point M0 is, in general, a cone of the
second order. For, let
be the equations of a curve C on the surface S. Then the three
functions f(£), $(£), \l/(t) satisfy the equation F(x, y, z) = 0, and
the first and second differentials satisfy the two relations
dF T cF cF
-^— dx + 7T- dy + -5— dz = 0,
ex cy cz
y cF , dF Y2) ()F cF dF
dx + — d,j + ~dz) +~d*x + ^-d2y + ~d*z = 0.
cy cz I ex Cy J cz
For the point x = x0, y = y0, z = z0 the first of these equations
reduces to an identity, and the second becomes
+ 2 r—=- dxdy + 2 ^—- dy dz + 2 £-f- dx dz = 0.
cx0dy0 0y0d*t dx0oz0
The equation of the locus of the tangents is given by eliminating
dx, dy, dz between the latter equation and the equation of a tangent
line
dx dy dz
which leads to the equation of a cone T of the second degree :
in,§M] SINGULAR POINTS MAXIMA AND MINIMA 115
(21)
c2F
(V — rV -I- ( Y — 11 V 4-
^A a%; -r- 8 , v,* y0;
2 — — — - (Ar — xn} ( Y —
3 o \ O/ \
c2F 2s F
2 =— =- C^- ?/o) (Z - «„) + 2 ^— ^~
On the other hand, applying Taylor's series with a remainder
and carrying the development to terms of the third order, the equa
tion of the surface becomes
(22) ^ 1.
7 , CF
where x, y, « in the ter)ns of the third order are to be replaced- by
x0 + 0(x-x0), y0 + 8(y-yd, z0 + 0(z-s0), respectively. The
equation of the cone T may be obtained by setting the terms of
the second degree in x — x0, y — y0, z — z0 in the equation (22) equal
to zero.
Let us then, first, suppose that the equation (21) represents a real
non-degenerate cone. Let the surface 5 and the cone T be cut by a
plane P which passes through two distinct generators G and G' of
the cone. In order to find the equation of the section of the sur
face 5 by this plane, let us imagine a transformation of coordinates
carried out which changes the plane P into a plane parallel to the
xy plane. It is then sufficient to substitute z = z<> in the equation (22).
It is evident that for this curve the point M0 is a double point with
real tangents ; from what we have just seen, this section is composed
of two branches tangent, respectively, to the two generators G, G'.
The surface S near the point M0 therefore resembles the two nappes
of a cone of the second degree near its vertex. Hence the point Mn
is called a conical point.
When the equation (21) represents an imaginary non-degenerate
cone, the point M0 is an isolated singular point of the surface fi.
Inside a sufficiently small sphere about such a point there exists no
set of solutions of the equation F(x, y, z) = 0 other than x = x0,
y = y0, z = z0. For, let M be a point in space near M0, p the
116 TAYLOR'S SERIES [m, §55
distance MM0, and a, (3, y the direction cosines of the line M0M.
Then if we substitute
X = X0 + pa, y = y0 + p0} K = Z0 + py,
the function F(x, y, 2) becomes
where L remains finite when p approaches zero. Since the equation
(21) represents an imaginary cone, the expression
)
or H ----- h 2
cannot vanish when the point (a, /?, y) describes the sphere
«2 + yS2 + y2 = 1.
Let w be a lower limit of the absolute value of this polynomial,
and let H be an upper limit of the absolute value of L near the
point Af0. If a sphere of radius m/H be drawn about M0 as center,
it is evident that the coefficient of p2 in the expression for F(x, y, z)
cannot vanish inside this sphere. Hence the equation
F(x, y,z) = 0
has no root except p = 0.
When the equation (21) represents two distinct real planes, two
nappes of the given surface pass through the point A/0, each of
which is tangent to one of the planes. Certain surfaces have a
line of double points, at each of which the tangent cone degenerates
into two planes. This line is a double curve on the surface along
which two distinct nappes cross each other. For example, the circle
whose equations are z = 0, x1 + y2 = 1 is a double line on the surface
whose equation is
«4 + 2z2(x* + ,f) - (r* + ,f- l)^ = 0.
When the equation (21) represents a system of two conjugate
imaginary planes or a double real plane, a special investigation is
necessary in each particular case to determine the form of the sur
face near the point M0. The above discussion will be renewed in
the paragraphs on extrema.
55. Extrema of functions of a single variable. Let the function f(x)
be continuous in the interval (a, 6), and let c be a point of that
Ill, §55] SINGULAR POINTS MAXIMA AND MINIMA 117
interval. The function /(#) is said to have an extremum (i.e. a
maximum or a minimum) for x = c if a positive number 77 can be
found such that the difference f(c -\- A) —f(c), which vanishes for
h = 0, has the same sign for all other values of h between — rj
and + i). If this difference is positive, the function f(x) has a
smaller value for x = c than for any value of x near c ; it is said
to have a minimum at that point. On the contrary, if the differ
ence f(c -f A) — /(c) is negative, the function is said to have a
maximum.
If the function f(x) possesses a derivative for x = c, that deriva
tive must vanish. For the two quotients
h -h
each of which approaches the limit /'(c) when h approaches zero, have
different signs ; hence their common limit /'(c) must be zero. Con
versely, let c be a root of the equation /'(#) = 0 which lies between
a and b, and let us suppose, for the sake of generality, that the
first derivative which does not vanish for x = c is that of order n,
and that this derivative is continuous when x = c. Then Taylor's
series with a remainder, if we stop with n terms, gives
which may be written in the form
/(« + A) -/(«) =
where c approaches zero with h. Let rj be a positive number such
that |/(n)(c) | is greater than e when x lies between c — 77 and c + 77.
For such values of x, /(n)(c) -f c has the same sign as f*-n)(c), and
consequently /(c -f A) —/(c) has the same sign as An/(n)(c). If
n is odd, it is clear that this difference changes sign with A, and
there is neither a maximum nor a minimum at x = c. If n is even,
f(c + A) — /(c) has the same sign as/(n)(c), whether A be positive
or negative ; hence the function is a maximum if /('°(c) is negative,
and a minimum if f(n)(c) is positive. It follows that the necessary
and sufficient condition that the function f(x) should have a maximum
or a minimum f or x = c is that the first derivative which does not
vanish for x = c should be of even order.
118 TAYLOR'S SERIES [III, §56
Geometrically, the preceding conditions mean that the tangent to
the curve y =f(x) at the point A whose abscissa is c must be par
allel to the axis of x, and moreover that the point A must not be
a point of inflection.
Notes. When the hypotheses which we have made are not satisfied
the function f(x) may have a maximum or a minimum, although
the derivative /'(#) does not vanish. If, for instance, the derivative
is infinite for x = c, the function will have a maximum or a mini
mum if the derivative changes sign. Thus the function y = a^ is at
a minimum for x = 0, and the corresponding curve has a cusp at the
origin, the tangent being the y axis.
When, as in the statement of the problem, the variable x is
restricted to values which lie between two limits a and b, it may
happen that the function has its absolute maxima and minima pre
cisely at these limiting points, although the derivative /'(x) does
not vanish there. Suppose, for instance, that we wished to find
the shortest distance from a point P whose coordinates are (a, 0)
to a circle C whose equation is xz + y2 — R2 — 0. Choosing for our
independent variable the abscissa of a point M of the circle C, we
find
d2 = PMZ = (x - a)2 + y2 = x2 + y2 - 2 ax + a2,
or, making use of the equation of the circle,
d2 = R2 + a* - 2 ax.
The general rule would lead us to try to find the roots of the derived
equation 2 a = 0, which is absurd. But the paradox is explained if
we observe that by the very nature of the problem the variable x
must lie between — R and + R. If a is positive, d2 has a minimum
for x = R and a maximum for x = — R.
56. Extrema of functions of two variables. Let f(x, y) be a con
tinuous function of x and y when the point M, whose coordinates
are x and y, lies inside a region ft bounded by a contour C. The
function f(x, y) is said to have an extremum at the point M0 (x0) ?/0)
of the region O if a positive number rj can be found such that the
difference
which vanishes for h = k = 0, keeps the same sign for all other sets
of values of the increments h and k which are each less than T in
Ill, §50] SINGULAR POINTS MAXIMA AND MINIMA 119
absolute value. Considering y for the moment as constant and
equal to ?/o, « becomes a function of the single variable x ; and, by
the above, the difference
cannot keep the same sign for small values of h unless the deriva
tive df/dx vanishes at the point M0. Likewise, the derivative df/dy
must vanish at MQ ; and it is apparent that the only possible sets of
values of x and y which can render the function f(x, y) an extre-
mum are to be found among the solutions of the two simultaneous
equations
*£=o, f£=o.
tix cy
Let x = x0, y = yQ be a set of solutions of these two equations.
We shall suppose that the second partial derivatives of f(x, y) do
not all vanish simultaneously at the point M0 whose coordinates
are (x0, y0), and that they, together with the third derivatives, are
all continuous near M0. Then we have, from Taylor's expansion,
A =
(23)
1.2
+ 6
(3)
We can foresee that the expression
will, in general, dominate the whole discussion.
In order that there be an extremum at MQ it is necessary and
sufficient that the difference A should have the same sign when the
point (XQ + h, y0 + k) lies anywhere inside a sufficiently small square
drawn about the point M0 as center, except at the center, where
A = 0. Hence A must also have the same sign when the point
(x0 + h, y0 + k) lies anywhere inside a sufficiently small circle whose
center is A/0 ; for such a square may always be replaced by its
inscribed circle, and conversely. Then let C be a circle of radius
r drawn about the point MQ as center. All the points inside this
circle are given by
120 TAYLOR'S SERIES [111, §56
where <£ is to vary from 0 to 2 TT, and p from — r to + r. We might,
indeed, restrict p to positive values, but it is better in what follows
not to introduce this restriction. Making this substitution, the
expression for A becomes
2 S
A = £• (A cos2<£ + 2 B sin <f> cos <f> + C sin2<£) + ^- Z,
where
and where Z is a function whose extended expression it would be
useless to write out, but which remains finite near the point (XQ, y0).
It now becomes necessary to distinguish several cases according to
the sign of B2 - A C.
First case. Let Bz — A C > 0. Then the equation
A cos2<£ + 2 B sin <£ cos <p + C sin2</> = 0
has two real roots in tan <£, and the first member is the difference
of two squares. Hence we may write
2 8
A — o" Ea (a cos ^ + b sin ^)2 "" P(a> cos ^ + b> sin ^)2] + ^ L>
where
a > 0, 0 > 0, aft' — fta' =jfc 0.
If <f> be given a value which satisfies the equation
a cos <£ + b sin <£ = 0,
A will be negative for sufficiently small values of p ; while, if <£ be
such that a'cos<f> + 6'sin<£ = 0, A will be positive for infinitesimal
values of p. Hence no number r can be found such that the differ
ence A has the same sign for any value of <£ when p is less than r.
It follows that the function f(x, y) has neither a maximum nor a
minimum for x = x0, y = y0.
Second case. Let B2 — A C < 0. The expression
A cos2</> + 2.Bcos
cannot vanish for any value of <£. Let m be a lower limit of its
absolute value, and, moreover, let H be an upper limit of the abso
lute value of the function L in a circle of radius R about (z0, y0) as
Ill, §57] SINGULAR POINTS MAXIMA AND MINIMA 121
center. Finally, let r denote a positive number less than R and less
than 3m/ H. Then inside a circle of radius r the difference A will
have the same sign as the coefficient of p2, i.e. the same sign as A
or C. Hence the function f(x, y~) has either a maximum or a mini
mum for x = XQ, y = ?/o-
To recapitulate, if at the point (x0, y0) we have
*'*">(>,
^dx0dy0
there is neither a maximum nor a minimum. But if
there is either a maximum or a minimum, depending on the sign of
the two derivatives c2f/dx%, o^f/dyl. There is a maximum if these
derivatives are negative, a minimum if they are positive.
57. The ambiguous case. The case where B2 — A C = 0 is not cov
ered by the preceding discussion. The geometrical interpretation
shows why there should be difficulty in this case. Let £ be the
surface represented by the equation z = f(x, ?/). If the function
f(x, y) has a maximum or a minimum at the point (XQ, y0), near
which the function and its derivatives are continuous, we must have
which shows that the tangent plane to the surface S at the point
M0, whose coordinates are (x0) y0, «0), must be parallel to the xy
plane. In order that there should be a maximum or a minimum it
is also necessary that the surface S, near the point M0, should lie
entirely on one side of the tangent plane ; hence we are led to study
the behavior of a surface with respect to its tangent plane near the
point of tangency.
Let us suppose that the point of tangency has been moved to the
origin and that the tangent plane is the xy plane. Then the equa
tion of the surface is of the form
(24) z = ax2 + 2 bxy + cy* + axs + 3 /3x*y + 3 yxy2 + Sy8,
where a, b, c are constants, and where a, /8, y, 8 are functions of x
and y which remain finite when x and y approach zero. This equa
tion is essentially the same as equation (19), where x0 and y0 have
been replaced by zeros, and h and k by x and y, respectively.
122
TAYLOR'S SERIES
[III, § 57
In order to see whether or not the surface S lies entirely on
one side of the xy plane near the origin, it is sufficient to study the
section of the surface by that plane. This section is given by the
equation
(25) ax* + 2bxy + cy2 + ax* + •• • = 0;
hence it has a double point at the origin of coordinates. If b2 — ac
is negative, the origin is an isolated double point (§ 53), and the
equation (25) has no solution except x = y = 0, when the point
(x, y) lies inside a circle C of sufficiently small radius r drawn
about the origin as center. The left-hand side of the equation (25)
keeps the same sign as long as the point (x, y) remains inside this
circle, and all the points of the surface S which project into the
interior of the circle C are on the same side of the xy plane except
the origin itself. In this case there is an extremum, and the por
tion of the surface S near the origin resembles a portion of a sphere
or an ellipsoid.
If b2 — ac> 0, the intersection of the surface S by its tangent
plane has two distinct branches Clf Cz which pass through the
origin, and the tangents to these two branches are given by the
equation
ax* + 2bxy + cy2 = 0.
Let the point (x, y) be allowed to move about in the neighborhood
of the origin. As it crosses either of the two branches Cx, C2, the
left-hand side of the equation (25) vanishes and changes sign.
Hence, assigning to each region of the plane in the neighborhood
of the origin the sign of the left-hand side of the equation (25), we
find a configuration similar to Fig. 7. Among the points of the
surface which project into points inside a circle about the origin in
the xy plane there are always some which
lie below and some which lie above the
xy plane, no matter how small the circle
be taken. The general aspect of the sur
face at this point with respect to its tan
gent plane resembles that of an imparted
hyperboloid or an hyperbolic paraboloid.
The function f(x, y) has neither a maxi-
mum nor a minimum at the origin.
The case where b2 — ac = 0 is the case in which the curve of
intersection of the surface by its tangent plane has a cusp at the
origin. We will postpone the detailed discussion of this case. If the
FlG- 7
Ill, §58] SINGULAR POINTS MAXIMA AND MINIMA 123
intersection is composed of two distinct branches through the origin,
there can be no extremum, for the surface again cuts the tangent
plane. If the origin is an isolated double point, the function f(x, y~)
has an extremum for x = y — 0. It may also happen that the inter
section of the surface with its tangent plane is composed of two
coincident branches. For example, the surface K — y* — 2 x*y -f x*
is tangent to the plane z = 0 all along the parabola y = x2. The
function ?/2 — 2 x2y -}- x4 is zero at every point on this parabola, but is
positive for all points near the origin which are not on the parabola.
58. In order to see which of these cases holds in a given example it is neces
sary to take into account the derivatives of the third and fourth orders, and some
times derivatives of still higher order. The following discussion, which is usually
sufficient in practice, is applicable only in the most general cases. When
62 — ac — 0 the equation of the surface may be written in the following form
by using Taylor's development to terms of the fourth order:
iW
(26) z - f(x, y) = A(xsinu -y cos w)2 + fa (x, y) + — --[x — + y — ,
24 \ dx dy /
ftr
Let us suppose, for definiteness, that A is positive. In order that the surface S
should lie entirely on one side of the xy plane near the origin, it is necessary that
all the curves of intersection of the surface by planes through the z axis should
lie on the same side of the xy plane near the origin. But if the surface be cut
by the secant plane
y = xtan 0,
the equation of the curve of intersection is found by making the substitution
x = p cos 0, y = p sin <j>
in the equation (26), the new axes being the old z axis and the trace of the secant
plane on the xy plane. Performing this operation, we find
z = A p2 (cos 0 sin w — cos w sin 0)2 -f K p3 + Lp*,
where K is independent of p. If tan w ^ tan 0, z is positive for sufficiently small
values of p ; hence all the corresponding sections lie above the xy plane near the
origin. Let us now cut the surface by the plane
y = x tan u.
If the corresponding value of K is not zero, the development of z is of the form
and changes sign with p. Hence the section of the surface by this plane has a
point of inflection at the origin and crosses the xy plane. It follows that the
function /(x, y) has neither a maximum nor a minimum at the origin. Such is
the case when the section of the surface by its tangent plane has a cusp of the
first kind, for instance, for the surface
z = w2 — x8.
124 TAYLOR'S SERIES [HI, §58
If K = 0 for the latter substitution, we would carry the development out to
terms of the fourth order, and we would obtain an expression of the form
where K\ is a constant which may be readily calculated from the derivatives of
the fourth order. We shall suppose that K\ is not zero. For infinitesimal val
ues of p, z has the same sign as K\ ; if K\ is negative, the section in question lies
beneath the xy plane near the origin, and again there is neither a maximum nor
a minimum. Such is the case, for example, for the surface z = y2 — x4, whose
intersection with the xy plane consists of the two parabolas y = ± x2. Hence,
unless K = 0 and K\ > 0 at the same time, it is evidently useless to carry the
investigation farther, for we may conclude at once that the surface crosses its
tangent plane near the origin.
But if K = 0 and KI > 0 at the same time, all the sections made by planes
through the z axis lie above the xy plane near the origin. But that does not
show conclusively that the surface does not cross its tangent plane, as is seen
by considering the particular surface
z - (y - x2) (y - 2 x2),
which cuts its tangent plane in two parabolas, one of which lies inside the other.
In order that the surface should not cross its tangent plane it is also necessary
that the section of the surface made by any cylinder whatever which passes
through the z axis should lie wholly above the xy plane. Let y = <f> (x) be the
equation of the trace of this cylinder upon the xy plane, where <f> (x) vanishes for
x = 0. The function F(x) =/[x, 0(x)] must be at a minimum for x = 0, what
ever be the function 0 (x). In order to simplify the calculation we will suppose
that the axes have been so chosen that the equation of the surface is of the form
z = Ay2 + <f>3(x, !/) + •••,
where A is positive. With this system of axes we have
=0 ^>0
8xo dyQ dx* dx0 dy0
at the origin.
The derivatives of the function F(x) are given by the formulae
F" x =
+ 3 £L $" (X) + 3 ^ W + ^ $'"(»),
dxdy dy2 cy
'+4 -*£-*' fart 4. 6 ^ *'*(x) | 1 ^ ^«(z) | g*^»'*
^3 f 53 f »3 f
6 -^- 0" (x) + 12 — ^- 0'0'' + 6 -^ <f>'24>"
dx2dy dxdy2 dy3
d2f ,„ d2f , ,„ //2 8/
dx dy dy2 dy
Ill, §59] SINGULAR POINTS MAXIMA AND MINIMA 125
from which, for x = y = 0, we obtain
c!/0
If tf>'(0) does not vanish, the function F(x) has a minimum, as is also apparent
from the previous discussion. But if <£'(()) = 0, we find the formulas
Hence, in order that F(x) be at a minimum, it is necessary that d*f/£x% vanish
and that the following quadratic form in tf>"(0),
r T~ 2 >
dx* cx20dy0 dy20
be positive for all values of 0"(0).
It is easy to show that these conditions are not satisfied for the above function
z = y2 — 3x2y + 2z4, but that they are satisfied for the function z = y2 + x*.
It is evident, in fact, that the latter surface lies entirely above the xy plane.
We shall not attempt to carry the discussion farther, for it requires extremely
nice reasoning to render it absolutely rigorous. The reader who wishes to exam
ine the subject in greater detail is referred to an important memoir by Ludwig
Scheffer, in Vol. XXXV of the Mathematische Annalen.
59. Functions of three variables. Let u = f(x, y, z) be a continuous
function of the three variables x, y, z. Then, as before, this func
tion is said to have an extremum (maximum or minimum) for a set
of values x0, y0, z0 if a positive number rj can be found so small
that the difference
which vanishes for h •= k = I = 0, has the same sign for all other
sets of values of h, k, I, each of which is less in absolute value
than i]. If only one of the variables *, y, z is given an increment,
while the other two are regarded as constants, we find, as above,
that u cannot be at an extremum unless the equations
are all satisfied, provided, of course, that these derivatives are con
tinuous near the point (or0, y0) z0~). Let us now suppose that x0, y0, z0
are a set of solutions of these equations, and let M0 be the point
whose coordinates are xw yQ, z0. There will be an extremum if a
sphere can be drawn about M0 so small that f(x, y, z) — f(x0) y0, z0)
126 TAYLOR'S SERIES [III, §59
has the same sign for all points (x, y, z) except M0 inside the sphere.
Let the coordinates of a neighboring point be represented by the
equations
x = x0 + pa, y = y0 + pft, z = z0 + py,
where a, ft, y satisfy the relation a2 + /32 + y2 = 1 ; and let us replace
x — x0, y — y0, z — z0 in Taylor's expansion of f(x, y, «) by pa, pft,
py, respectively. This gives the following expression for A :
A = p2[>O, ft, y) +.--],
where <f>(a, ft, y) denotes a quadratic form in a, ft, y whose coeffi
cients are the second derivatives of f(x, y, z), and where Z is a
function which remains finite near the point MQ. The quadratic
form may be expressed as the sum of the squares of three distinct
linear functions of a, ft, y, say P, P', P", multiplied by certain con
stant factors a, a', a", except in the particular case when the dis
criminant of the form is zero. Hence we may write, in general,
*(a, ft, y) = aP2 + a'P'2 + a"P"2,
where a, a', a" are all different from zero. If the coefficients a, a', a"
have the same sign, the absolute value of the quadratic form <f> will
remain greater than a certain lower limit when the point a, ft, y
describes the sphere
«2 + ft2 + y2 = 1,
and accordingly A has the same sign as a, a', a" when p is less than
a certain number. Hence the f imction f(x, y, z) has an extremum.
If the three coefficients a, a', a" do not all have the same sign,
there will be neither a maximum nor a minimum. Suppose, for
example, that a > 0, a' < 0, and let us take values of a, ft, y which
satisfy the equations P' = 0, P" = 0. These values cannot cause P
to vanish, and A will be positive for small values of p. But if, on
the other hand, values be taken for a, ft, y which satisfy the equa
tions P = 0, P" = 0, A will be negative for small values of p.
The method is the same for any number of independent variables :
the discussion of a certain quadratic form always plays the prin
cipal role. In the case of a function u = f(x, y, z) of only three
independent variables it may be noticed that the discussion is
equivalent to the discussion of the nature of a surface near a singu
lar point. For consider a surface 2 whose equation is
F(*> y, «) =f(x> y> *) -f(*o, y0, O = 0;
Ill, §60] SINGULAR POINTS MAXIMA AND MINIMA 127
this surface evidently passes through the point Mn whose coordi
nates are (x0, y0, «0), and if the function f(x, y, z) has an extremum
there, the point M0 is a singular point of 2- Hence, if the cone of
tangents at M0 is imaginary, it is clear that F(x, y, z) will keep the
same sign inside a sufficiently small sphere about M0 as center, and
/(#» 2/j «) wiH surely have a maximum or a minimum. But if the
cone of tangents is real, or is composed of two real distinct planes,
several nappes of the surface pass through A/0, and F(x, y, z)
changes sign as the point (x, y, z) crosses one of these nappes.
60. Distance from a point to a surface. Let us try to find the maximum and the
minimum values of the distance from a fixed point (a, b, c) to a surface S whose
equation is F(x, y, z) = 0. The square of this distance,
u = d* = (x - a)2 + (y - 6)» + (• - c)»,
is a function of two independent variables only, — x and y, for example, if z be
considered as a function of x and y defined by the equation F = 0. In order
that u be at an extremum for a point (x, y, z) of the surface, we must have, for
the coordinates of that point,
1 du , dz
= (x — a) + (z — c) — = 0,
2 dx 'ix
1 du . dz
_ = y _ & + (z - c) — = 0.
2 dy cy
We find, in addition, from the equation F = 0, the relations
dF dFdz dF dFdz rt
-\ = U, 1 = U,
dx dz dx dy dz dy
whence the preceding equations take the form
x — a _ y — b _ z — c
djr ~~ c_F_ " d_F_ '
dx dy dz
This shows that the normal to the surface S at the point (x, y, z) passes through
the point (a, 6, c). Hence, omitting the singular points of the surface S, the
points sought for are the feet of normals let fall from the point (a, 6, c) upon the
surface S. In order to see whether such a point actually corresponds to a maxi
mum or to a minimum, let us take the point as origin and the tangent plane as
the xy plane, so that the given point shall lie upon the axis of z. Then the func
tion to be studied has the form
u = x* + y2 + (z - c)2,
where z is a function of x and y which, together with both its first derivatives,
vanishes for x = y = 0. Denoting the second partial derivatives of z by r, s, t,
we have, at the origin,
^ = 2(1 -or), -fiL — 1* ^ = 2(l-cO,
dx* dxdy dy2
128 TAYLOR'S SERIES [III, §61
and it only remains to study the polynomial
A(C) = C2«2 _ (1 _ cr) (1 _ ct) = C2(S2 _ rt) + (r + t)c _ L
The roots of the equation A (c) = 0 are always real by virtue of the identity
(r + £)2 + 4 (s2 — rt) = 4 s2 + (r — t)2. There are now several cases which must
be distinguished according to the sign of s2 — rt.
First case. Let s2 — rt < 0. The two roots Ci and c2 of the equation A (c) = 0
have the same sign, and we may write A(c) = (s2 — rt) (c — Ci) (c — Cj). Let us
now mark the two points A\ and A% of the z axis whose coordinates are c\ and c2.
These two points lie on the same side of the origin ; and if we suppose, as is
always allowable, that r and t are positive, they lie on the positive part of the
z axis. If the given point A (0, 0, c) lies outside the segment AiAz, A(c) is
negative, and the distance OA is a maximum or a minimum. In order to see
which of the two it is we must consider the sign of 1 — cr. This coefficient
does not vanish except when c = 1 /r ; and this value of c lies between Ci and c2,
since A (1/r) = s2/r2. But, for c = 0, 1 — cr is positive ; hence 1 — cr is posi
tive, and the distance OA is a minimum if the point A and the origin lie on
the same side of the segment A\A%. On the other hand, the distance OA i& a
maximum if the point A and the origin lie on different sides of that segment.
When the point A lies between AI and A2 the distance is neither a minimum
nor a maximum. The case where A lies at one of the points AI, A2 is left in
doubt.
Second case. Let s2 — rt > 0. One of the two roots c\ and c2 of A (c) = 0 is
positive and the other is negative, and the origin lies between the two points
A\ and J.a. If the point A does not lie between A\ and A2, A(c) is positive
and there is neither a maximum nor a minimum. If A lies between AI and
A 2, A (c) is negative, 1 — cr is positive, and hence the distance OA is a minimum.
Third case. Let s2 - rt = 0. Then A(c) = (r + t) (c - cj), and it is easily
seen, as above, that the distance OA is a minimum if the point A and the origin
lie on the same side of the point AI, whose coordinates are (0, 0, Ci), and that
there is neither a maximum nor a minimum if the point AI lies between the point
A and the origin.
The points AI and A2 are of fundamental importance in the study of curva
ture ; they are the principal centers of curvature of the surface S at the point 0.
61. Maxima and minima of implicit functions. We often need to find
the maxima and minima of a function of several variables which
are connected by one or more relations. Let us consider, for
example, a function to = f(x, y, z, tt) of the four variables x, y, z, u,
which themselves satisfy the two equations
/i (*, y, *, «) = 0, /,(*, y, z, M) = 0.
For definiteness, let us think of x and y as the independent vari
ables, and of z and u as functions of x and y defined by these equa
tions. Then the necessary conditions that u> have an extremum are
HI, §61] SINGULAR POINTS MAXIMA AND MINIMA 129
2£ + ^ + ^ = 0 !/+££?£ + 3£?!? = 0
dx dz dx du dx dy dz dy du dy
and the partial derivatives dz/dx, du/dx, dz/dy, du/dy are given
by the relations
M_i.^^4.M^ = o 2£ 4. ££ a * 4. §£ ?!f = o
dx dz dx^ du dx dx dz dx du dx
0/i ,0/10*, 0/i0"_n 0/, , 0/,0* , 0/,0u
"^ -- P "TT~ o -- P "5 — ~^T~ — ", "a P "o~ o P a "a~ — U-
^z/ ^* dy du dy cy oz cy cu cy
The elimination of dz/dx, du/dx, dz/dy, du/dy leads to the new
equations of condition
.p« _ft
/)(*, *, u) '
which, together with the relations /x = 0, /2 = 0, determine the val
ues of x, y, z, u, which may correspond to extrema. But the equa
tions (27) express the condition that we can find values of \ and p.
which satisfy the equations
- -t- A -^- -I- w. IT- = "j a l~A"o •" I* "a~ = wi
c OX OX dy dy oy
3£ dJl ^2 = o ^ + X^4 ^2 = 0-
02 3s ds dw du du
hence the two equations (27) may be replaced by the four equations
(28), where X and p. are unknown auxiliary functions.
The proof of the general theorem is self-evident, and we may
state the following practical rule :
Given a function
Atm /m . *F \ *
iCjj ^2, , *,n)
of n variables, connected by h distinct relations
in order to find the values of x^ a;2, •••, xn which may render this
function an extremum we must equate to zero the partial derivatives
of the auxiliary function
regarding \1} X2, • • • , XA as constants.
130 TAYLOR'S SERIES [III, §62
62. Another example. We shall now take up another example, where the mini
mum is not necessarily given by equating the partial derivatives to zero. Given
a triangle ABC; let us try to find a point P of the plane for which the sum
PA + PB + PC of the distances from P to the vertices of the triangle is a
minimum. Let (01, 61), (a2, 62), (a3, 63) be respectively the coordinates of the
vertices A, B, C referred to a system of rectangular coordinates. Then the func
tion whose minimum is sought is
(29) z = V(x - oi)a + (y- 6i)a + V(x - a2)2 + (y - 62)2 + V(x - a,)* + (y -
where each of the three radicals is to be taken with the positive sign. This equa
tion (29) represents a surface S which is evidently entirely above the xy plane,
and the whole question reduces to that of finding the point on this surface which
is nearest the xy plane. From the relation (29) we find
x — a-2 x — as
H -- , +
s* V(x - m)a + (y - &!)2 V(x - a2)2 + (y - 62)2 V(z - as)2 + (y -
dz _ y -bi y - 62 y -b3
1
ft V(z _ ai)2 + (y - &!)2 V(x - as)2 + (y - btf V(x - a3)2 + (y - 63)2 '
and it is evident that these derivatives are continuous, except in the neighbor
hood of the points A, B, C, where they become indeterminate. The surface S,
therefore, has three singular points which project into the vertices of the given
triangle. The minimum of z is given by a point on the surface where the tan
gent plane is parallel to the xy plane, or else by one of these singular points. In
order to solve the equations cz/cx = 0, cz/dy = 0, let us write them in the
form
x — 0,1 x — a2 x — a3
— i : —
V(x - ai)» + (y- &i)2 V(x - a2)2 + (y - 62)2 V(x - as)2+ (y -
y-E>i y - &2 _ y - b3
V(X - fll)2 + (y - &!)» V(X - aa)2 +(y- 62)2
Then squaring and adding, we find the condition
V(x - a2)2 + (y -
_0
The geometrical interpretation of this result is easy : denoting by a and /3 the
cosines of the angles which the direction PA makes with the axes of x and j/,
respectively, and by a' and /3' the cosines of the angles which PB makes with the
same axes, we may write this last condition in the form
1 + 2 (aa' + flS7) = 0,
or, denoting the angle APE by o>,
2 cos u + 1 = 0.
Hence the condition in question expresses that the segment AB subtends an
angle of 120° at the point P. For the same reason each of the angles BPC and
CPA must be 120°.* It is clear that the point P must lie inside the triangle
* The reader is urged to draw the figure.
Ill, §63] SINGULAR POINTS MAXIMA AND MINIMA 131
ABC, and that there is no point which possesses the required property if any
angle of the triangle ABC is equal to or greater than 120°. In case none of the
angles is as great as 120°, the point P is uniquely determined by an easy con
struction, as the intersection of two circles. In this case the minimum is given
by the point P or by one of the vertices of the triangle. But it is easy to show
that the sum PA + PB + PC is less than the sum of two of the sides of the tri
angle. For, since the angles APB and APC are each 120°, we find, from the
two triangles PAG and PBA, the formulae
AB = V«i + b'2 + ab, AC = Va2 + c2 + oc,
where PA = a, PB = 6, PC = c. But it is evident that
Vo2 + 62 + 06 > 6+-, Va'2 -)- c2 + ac> c + ~,
2 2
and hence
AB + AC > a + b + c.
The point P therefore actually corresponds to a minimum.
When one of the angles of the triangle ABC is equal to or greater than 120°
there exists no point at which each of the sides of the triangle ABC subtends an
angle of 120°, and hence the surface S has no tangent plane which is parallel to
the xy plane. In this case the minimum must be given by one of the vertices of
the triangle, and it is evident, in fact, that this is the vertex of the obtuse angle.
It is easy to verify this fact geometrically.
63. D'Alembert's theorem. Let F(x, y) be a polynomial in the two variables
x and y arranged into homogeneous groups of ascending order
F(x, y) = H + <f>P (x, y) + <t>p + i (x, y) + ••• + 0m(x, y),
where H is a constant. If the equation <f>p (x, y) = 0, considered as an equation
in y/Xj has a simple root, the function F(x, y) cannot have a maximum or a mini
mum for x = y = 0. For it results from the discussion above that there exist sec
tions of the surface z + H = F(x, y) made by planes through the z axis, some
of which lie above the xy plane and others below it near the origin. From this
remark a demonstration of d'Alembert's theorem may be deduced. For, let/(z)
be an integral polynomial of degree m,
/(z) = A9 + AIZ + A2z* + ••• + Amzm,
where the coefficients are entirely arbitrary. In order to separate the real and
imaginary parts let us write this in the form
f(x + iy) = a0 + ib0 + (ai + t&i) (x + iy) + • • • + (am + ibm) (x + iy)m,
where OQ, &o» «i, &ii • • •» «m, &m are real. We have then
f(z) = P+iQ,
where P and Q have the following meanings :
P = o0 + etix - biy -\ ---- ,
Q = &0 + bix + a^y + • • • ;
and hence, finally,
132 TAYLOR'S SERIES [III, §63
We will first show that |/(z) |, or, what amounts to the same thing, that
ps -j. Q2f cannot be at a minimum for z = y = 0 except when a0 = 60 = 0. For
this purpose we shall introduce polar coordinates p and 0, and we shall suppose,
for the sake of generality, that the first coefficient after A0 which does not
vanish is Ap. Then we may write the equations
P = o0 + (ap cos p<f> — bp sin p<f>) pp + ••'•,
Q = b0 + (bp cos p0 + dp sin p<j>) pp + • • • ,
P2 + Q2 = of, + 6g + 2/>p [(aoOp + b0bp) cosptf. + (b0ap - a0bp) sinp0] + • • • ,
where the terms not written down are of degree higher than p with respect to p.
But the equation
(aoap + b0bp) cosp<j> + (b0ap — a0bp) smp<J> = 0
gives tan p$ = K, which determines p straight lines which are separated by
angles each equal to 2 n /p. It is therefore impossible by the above remark that
P2 + Q2 should have a minimum for z = y = 0 unless the quantities
aoap -f &o&p» boa,p — (tobp
both vanish. But, since a2 + ft2 is not zero, this would require that a0 = 60 = 0 ;
that is, that the real and the imaginary parts of /(z) should both vanish at the
origin.
If |/(z) | has a minimum for z = a, y — /3, the discussion may be reduced to
the preceding by setting z = a + i/3 + z'. It follows that \f(z) \ cannot be at a
minimum unless P and Q vanish separately for x= a, y = p.
The absolute value of /(z) must pass through a minimum for at least one
value of z, for it increases indefinitely as the absolute value of z increases indefi
nitely. In fact, we have
where the terms omitted are of degree less than 2 m in p. This equation may be
written in the form
where t approaches zero as p increases indefinitely. Hence a circle may be
drawn whose radius R is so large that the value of VP2 + Q2 is greater at every
point of the circumference than it is at the origin, for example. It follows that
there is at least one point
x = a, y = £
inside this circle for which Vp« + Q2 is at a minimum. By the above it fol
lows that the point x = a, y = /3 is a point of intersection of the two curves
P = 0, Q = 0, which amounts to saying that z = a + /3i is a root of the equation
/(*)=«.
In this example, as in the preceding, we have assumed that a function of the
two variables x and y which is continuous in the interior of a limited region
actually assumes a minimum value inside or on the boundary of that region.
This is a statement which will be readily granted, and, moreover, it will be
rigorously demonstrated a little later (Chapter VI).
Ill, EM.] EXERCISES 133
EXERCISES
1. Show that the number 0, which occurs in Lagrange's form of the re
mainder, approaches the limit l/(n + 2) as A approaches zero, provided that
/(» + 2J(a) is not zero.
2. Let F(x) be a determinant of order n, all of whose elements are functions
of x. Show that the derivative F'(x) is the sum of the n determinants obtained
by replacing, successively, all of the elements of a single line by their deriva
tives. State the corresponding theorem for derivatives of higher order.
3. Find the maximum and the minimum values of the distance from a fixed
point to a plane or a skew curve ; between two variable points on two curves ;
between two variable points on two surfaces.
4. The points of a surface S for which the sum of the squares of the dis
tances from n fixed points is an extremum are the feet of the normals let fall
upon the surface from the center of mean distances of the given n fixed points.
5. Of all the quadrilaterals which can be formed from four given sides, that
which is inscriptible in a circle has the greatest area. State the analogous
theorem for polygons of n sides.
6. Find the maximum volume of a rectangular parallelepiped inscribed in
an ellipsoid.
7. Find the axes of a central quadric from the consideration that the vertices
are the points from which the distance to the center is an extremum.
8. Solve the analogous problem for the axes of a central section of an ellipsoid.
9. Find the ellipse of minimum area which passes through the three vertices
of a given triangle, and the ellipsoid of minimum volume which passes through
the four vertices of a given tetrahedron.
10. Find the point from which the sum of the distances to two given straight
lines and the distance to a given point is a minimum.
[JOSEPH BERTRAND.]
11. Prove the following formulae :
log (3 + 2) = 2 log(z + 1) - 2 log (x-l) + log(x - 2)
—- — _— -
z3-3z 3Vx3-3z/ 6\z3-3z
[BORDA'S Series.]
log(x + 6) = log(x -f 4) + log(x + 3) - 2 logx
+ log(z - 3) + log(z - 4) - log(x - 5)
of 72 If 72 y -1
|_x* - 25z2 + 72 3 \z* - 26z2 + 72/ J
[HARO'S Series.]
CHAPTER IV
DEFINITE INTEGRALS
I. SPECIAL METHODS OF QUADRATURE
64. Quadrature of the parabola. The determination of the area
bounded by a plane curve is a problem which has always engaged
the genius of geometricians. Among the examples which have
come down to us from the ancients one of the most celebrated is
Archimedes' quadrature of the parabola. We shall proceed to
indicate his method.
Let us try to find the area bounded by the arc A CB of a parabola
and the chord A 13. Draw the diameter CD, joining the middle
point D of AB to the point C, where the tangent is parallel to AB.
Connect AC and BC, and let E and E' be the points where the
tangent is parallel to .BC and
AC, respectively. We shall
first compare the area of the
triangle BEC, for instance,
with that of the triangle ABC.
Draw the tangent ET, which
cuts CD at T. Draw the diam
eter EF, which cuts CB at F;
and, finally, draw EK and FH
parallel to the chord AB. By
an elementary property of the
parabola TC = CK. Moreover,
CT — EF = KH, and hence
EF= CH/2 = CD/ 4. The
areas of the two triangles BCE
and BCD, since they have the
same base BC, are to each other as their altitudes, or as EF is
to CD. Hence the area of the triangle BCE is one fourth the area
of the triangle BCD, or one eighth of the area 5 of the triangle ABC.
The area of the triangle A CE' is evidently the same. Carrying out
the same process upon each of the chords BE, CE, CE', E'A, we
134
FIG. 8
IV, §65]
SPECIAL METHODS
135
obtain four new triangles, the area of each of which is S/82, and so
forth. The nih operation gives rise to 2" triangles, each having the
area S/8n. The area of the segment of the parabola is evidently
the limit approached by the sum of the areas of all these triangles
as n increases indefinitely ; that is, the sum of the following descend
ing geometrical progression :
and this sum is 4 5/3. It follows that the required area is equal to
two thirds of the area of a parallelogram whose sides are AB and CD.
Although this method possesses admirable ingenuity, it must be
admitted that its success depends essentially upon certain special
properties of the parabola, and that it is lacking in generality. The
other examples of quadratures which we might quote from ancient
writers would only go to corroborate this remark : each new curve
required some new device. But whatever the device, the area to be
evaluated was always split up into elements the number of which
was made to increase indefinitely, and it was necessary to evaluate
the limit of the sum of these partial areas. Omitting any further
particular cases,* we will proceed at once to give a general method
of subdivision, which will lead us naturally to the Integral Calculus.
65. General method. For the sake of definiteness, let us try to
evaluate the area 5 bounded by a curvilinear arc A MB, an axis xx'
which does not cut that arc, and two perpendiculars AA0 and BB0 let
fall upon xx' from
the points A and B.
We will suppose
further that a par
allel to these lines
AA0, BB0 cannot
cut the arc in more
than one point, as
indicated in Fig. 9.
Let us divide the segment A0B0 into a certain number of equal or
unequal parts by the points Pl9 P2, •••, Pn.\, and through these
points let us draw lines PiQi, P2Q2, •••, Pn_iQH_i parallel to AA0
and meeting the arc AB in the points Qi, Q2> •• • j Qn-i> respectively.
FIG. 9
* A large number of examples of determinations of areas, arcs, and volumes by
the methods of ancient writers are to be found in Duhamel's TraiM.
136 DEFINITE INTEGRALS [IV, §65
Now draw through A a line parallel to xx', cutting Pt Qt at q± ;
through Qi a parallel to xx', cutting P2Q2 at q2; and so on. We
obtain in this way a sequence of rectangles RI, R2, • • • , Rt, • • • , Rn.
Each of these rectangles may lie entirely inside the contour A£B0A0,
but some of them may lie partially outside that contour, as is
indicated in the figure.
Let a( denote the area of the rectangle R{, and /^ the area bounded
by the contour Pi_lPiQiQi_l. In the first place, each of the ratios
fii/aD ^2/a2> ••••> A/ai> '•• approaches unity as the number of
points of division increases indefinitely, if at the same time each
of the distances A0Pl} PiP2, •••, P^P,, ••• approaches zero. For
the ratio /?,/<*,, for example, evidently lies between £,-/ '/*,-_! Q,-_i and
Li/Pi_lQi_l, where lf and L{ are respectively the minimum and the
maximum distances from a point of the arc Qi_lQi to the axis xx'.
But it is clear that these two fractions each approach unity as the
distance Pt_lPi approaches zero. It therefore follows that the ratio
a-! + or2 H ----- h an
A + & + • • ' + &
which lies between the largest and the least of the ratios tfi//3i,
a2//?2> •••> an/ Pm wiH a^so approach unity as the number of the
rectangles is thus indefinitely increased. But the denominator of
this ratio is constant and is equal to the required area S. Hence
this area is also equal to the limit of the sum ax + a2 + • • • + an, as
the number of rectangles n is indefinitely increased in the manner
specified above.
In order to deduce from this result an analytical expression for
the area, let the curve AB be referred to a system of rectangular
axes, the x axis Ox coinciding with xx', and let y =f(x) be the
equation of the curve AB. The function f(x) is, by hypothesis, a
continuous function of x between the limits a and b, the abscissae
of the points A and B. Denoting by x1} x2, • • • , xn_l the abscissas
of the points of division P1} P2, •••, Pn_j, the bases of the above
rectangles are x± — a, x.2 — xly • • • , xt — x^^ • • • , b — xn_l, and their
altitudes are, in like manner, f(a)tf(x^t ••-, /(«<_,), •••, /(»»_i).
Hence the area S is equal to the limit of the following- sum :
(1) (Xl - a)f(a) + (x2 - *,)/(*!) + - - • + (b - *„_,)/(*„_,),
as the number n increases indefinitely in such a way that each of
the differences xl — a, x2 — xl} ••• approaches zero.
SPECIAL METHODS 137
66. Examples. If the base AB be divided into n equal parts, each
of length h (b — a = nh), all the rectangles have the same base h,
and their altitudes are, respectively,
/(a), /(a + h), f(a + 2 A), • • -, /[« + (» - 1) A].
It only remains to find the limit of the sum
h !/(«) +/(« + 70 +/(« + 2 70 + •
where
as the integer w increases indefinitely. This calculation becomes
easy if we know how to find the sum of a set of values f(x) corre
sponding to a set of values of x which form an arithmetic progres
sion ; such is the case if f(x) is simply an integral power of x, or,
again, ift/(o;)= s'mmx or /"(#)= cosmx, etc.
Let us reconsider, for example, the parabola x* = 2py, and let us try
to find the area enclosed by an arc OA of this parabola, the axis of x,
and the straight line x = a which passes through the extremity A.
The length being divided into n equal parts of length h (nh = a), we
must try to find by the above the limit of the sum
The quantity inside the parenthesis is the sum of the squares of the
first (n — 1) integers, that is, n(n — 1) (2 n — l)/6; and hence the
foregoing sum is equal to
As n increases indefinitely this sum evidently approaches the limit
a*/6p = (1/3) (a. a2/2p), or one third of the rectangle constructed
upon the two coordinates of the point A, which is in harmony with
the result found above.
In other cases, as in the following example, which is due to
Fermat, it is better to choose as points of division points whose
abscissae are in geometric progression.
Let us try to find the area enclosed by the curve y = Ax*, the
axis of x, and the two straight lines x — a, x = I (0 < a < b), where
138 DEFINITE INTEGRALS [IV, §66
the exponent /* is arbitrary. In order to do so let us insert between
a and b, n — 1 geometric means so as to obtain the sequence
where the number a satisfies the condition a (1 4- a)" = b. Tak
ing this set of numbers as the abscissae of the points of division, the
corresponding ordinates have, respectively, the following values :
ay, Aa* (I + a)2*1, ••
and the area of the pth rectangle is
[a (1 + a)" — a (1 + a)*"1] Aa*(l + a) <»-»* = Aa»+l
Hence the sum of the areas of all the rectangles is
If /i + 1 is not zero, as we shall suppose first, the sum inside the
parenthesis is equal to
or, replacing a (1 + a)" by i, the original sum may be written in the
form
\^ I .v
As a approaches zero the quotient [(1 + a)M + 1 — !]/<* approaches
as its limit the derivative of (1 + a)^ + 1 with respect to a for a = 0,
that is, /i + 1 ; hence the required area is
If p. = — 1, this calculation no longer applies. The sum of the
areas of the inscribed rectangles is equal to nAa, and we have to
find the limit of the product na where n and a are connected by the
relation
a(l -I- a)" = b.
From this it follows that
, b a . b 1
na = log - r — = log -
&alog(l + «) ha
, ,4
log(l
IV, § 67]
SPECIAL .METHODS
139
where the symbol log denotes the Naperian logarithm. As a
approaches zero, (1 + a)1/* approaches the number e, and the prod
uct na approaches log (b fa). Hence the required area is equal to
vl log (&/«)•
67. Primitive functions. The invention of the Integral Calculus
reduced the problem of evaluating a plane area to the problem of
finding a function whose derivative is known. Let y =f(xj be the
equation of a curve referred to two rectangular axes, where the
function f(x) is continuous. Let us consider the area enclosed by
this curve, the axis of x, a fixed ordinate M0P0, and a variable
ordinate MP, as a function of the abscissa x of the variable ordinate.
In order to include all pos
sible cases let us agree to
denote by A the sum of the
areas enclosed by the given
curve, the x axis, and the
straight lines M0P0, MP,
each of the portions of
this area being affected
by a certain sign : the
sign + for the portions to
the right of M0P0 and above Ox, the sign
right of M0P0 and below Ox, and the opposite convention for por
tions to the left of M^Pg. Thus, if MP were in the position M'P', we
would take A equal to the difference
JI/0P0C - M'P'C;
and likewise, if MP were at M"P", A = M"P"D - M0P0D.
With these conventions we shall now show that the derivative of
the continuous function A, defined in this way, is precisely /(#). As
in the figure, let us take two neighboring ordinates MP, NQ, whose
abscissae are x and x -f Ax. The increment of the area A.4 evidently
lies between the areas of the two rectangles which have the same
base PQ, and whose altitudes are, respectively, the greatest and the
least ordinates of the arc M N. Denoting the maximum ordinate by
H and the minimum by h, we may therefore write
AAz < <\A < 7/Ax,
or, dividing by Ax, h < A/l /Ax < //. As Ax approaches zero, // and
h approach the same limit MP, or /(x), since /(x) is continuous.
FIG. 10
for the portions to the
140 DEFINITE INTEGRALS [IV, §<;8
Hence the derivative of A is f(x). The proof that the same result
holds for any position of the point .17 is left to the reader.
If we already know a primitive function of f(x), that is, a function
F(x) whose derivative is/(z), the difference A — F(x) is a constant,
since its derivative is zero (§ 8). In order to determine this con
stant, we need only notice that the area A is zero for the abscissa
x = a of the line MP. Hence
A =F(x)-F(a).
It follows from the above reasoning, first, that the determination
of a plane area may be reduced to the discovery of a primitive func
tion; and, secondly (and this is of far greater importance for us),
that every continuous function f(x) is the derivative of some other
function. This fundamental theorem is proved here by means of
a somewhat vague geometrical concept, — that of the area under a
plane curve. This demonstration was regarded as satisfactory for a
long time, but it can no longer be accepted. In order to have a stable
foundation for the Integral Calculus it is imperative that this theo
rem should be given a purely analytic demonstration which does not
rely upon any geometrical intuition whatever. In giving the above
geometrical proof the motive was not wholly its historical interest,
however, for it furnishes us with the essential analytic argument of
the new proof. It is, in fact, the study of precisely such sums as
(1) and sums of a slightly more general character which will be
of preponderant importance. Before taking up this study we must
first consider certain questions regarding the general properties of
functions and in particular of continuous functions.*
II. DEFINITE INTEGRALS ALLIED GEOMETRICAL CONCEPTS
68. Upper and lower limits. An assemblage of numbers is said to
have an upper limit (see ftn., p. 91) if there exists a number N so
large that no member of the assemblage exceeds N. Likewise, an
assemblage is said to have a lower limit if a number N' exists than
which no member of the assemblage is smaller. Thus the assem
blage of all positive integers has a lower limit, but no upper limit ;
* Among the most important works on the general notion of the definite integral
there should be mentioned the memoir by Riemann : fiber die Mb'glichkeit, eine Func
tion durch eine trigonometrische Reihe darzustellen (Werke, 2d ed., Leipzig, 1892,
p. 239 ; and also French translation by Laugel, p. 225) ; and the memoir by Darboux, to
which we have already referred : Sur les fonctions discontinues (Annales de VEcole
Normals Suptrieure, 2d series, Vol. IV).
IV, §68] ALLIED GEOMETRICAL CONCEPTS 141
the assemblage of all integers, positive and negative, has neither ;
and the assemblage of all rational numbers between 0 and 1 has
both a lower and an upper limit.
Let (E) be an assemblage which has an upper limit. With
respect to this assemblage all numbers may be divided into two
classes. We shall say that a number a belongs to the first class if
there are members of the assemblage (7?) which are greater than a,
and that it belongs to the second class if there is no member of the
assemblage (7?) greater than a. Since the assemblage (7?) has an
upper limit, it is clear that numbers of each class exist. If A be
a number of the first class and B a number of the second class, it
is evident that A < B ; there exist members of the assemblage (7?)
which lie between A and B, but there is no member of the assem
blage (7?) which is greater than B. The number C = (A -f jB)/2
may belong to the first or to the second class. In the former case
we should replace the interval (A, B*) by the interval (C, 7?), in the
latter case by the interval (A, C). The new interval (.4^ 7^) is half
the interval (^4, B) and has the same properties : there exists at least
one member of the assemblage (7£) which is greater than A1} bnt none
which is greater than B±. Operating upon (Al} B^) in the same way
that we operated upon (A, B}, and so on indefinitely, we obtain an
unlimited sequence of intervals (A, 73), (Alf 7^), (A2, 7?2), •— j each
of which is half the preceding and possesses the same property
as (A, B} with respect to the assemblage (/?). Since the numbers
A, AI, Az, •••, An never decrease and are always less than B, they
approach a limit A (§ 1). Likewise, since the numbers B, B1} B2, • • •
never increase and are always greater than A, they approach a limit X'.
Moreover, since the difference Bn — An = (B — A ) /2" approaches zero
as n increases indefinitely, these limits must be equal, i.e. A' = A.
Let L be this common limit ; then L is called the iqjper limit of the
assemblage (7?). From the manner in which we have obtained it,
it is clear that L has the following two properties :
1) No member of the assemblage (7i) is greater than L.
2) There always exists a member of the assemblage (7?) which is
greater than L — e, where c is any arbitrarily small positive number.
For let us suppose that there were a member of the assemblage
greater than L, say L + h (Ji > 0). Since Bn approaches L as n
increases indefinitely, Bn will be less than L -f h after a certain
value of n. But this is impossible since Bn is of the second class.
On the other hand, let e be any positive number. Then, after a
142 DEFINITE INTEGRALS [IV, §69
certain value of n, An will be greater than L — e ; and since there are
members of (E) greater than An, these numbers will also be greater
than L — e. It is evident that the two properties stated above can
not apply to any other number than L.
The upper limit may or may not belong to the assemblage (£).
In the assemblage of all rational numbers which do not exceed 2,
for instance, the number 2 is precisely the upper limit, and it belongs
to the assemblage. On the other hand, the assemblage of all irra
tional numbers which do not exceed 2 has the upper limit 2, but
this upper limit is not a member of the assemblage. It should be
particularly noted that if the upper limit L does not belong to the
assemblage, there are always an infinite number of members of (£)
which are greater than L — c, no matter how small e be taken. For if
there were only a finite number, the upper limit would be the largest
of these and not L. When the assemblage consists of n different
numbers the upper limit is simply the largest of these n numbers.
It may be shown in like manner that there exists a number L\
in case the assemblage has a lower limit, which has the following
two properties :
1) No member of the assemblage is less than L'.
2) There exists a member of the assemblage which is less than
L' -\- (., where e is an arbitrary positive number.*
This number L' is called the lower limit of the assemblage.
69. Oscillation. Let/(«) be a function of x defined in the closed f
interval (a, £) ; that is, to each value of x between a and b and to each
of the limits a and b themselves there corresponds a uniquely deter
mined value of f(x}. The function is said to be finite in this closed
interval if all the values which it assumes lie between two fixed
numbers A and B. Then the assemblage of values of the function
has an upper and a lower limit. Let M and m be the upper and
lower limits of this assemblage, respectively ; then the difference
* Whenever all numbers can be separated into two classes A and B, according to
any characteristic property, in such a way that any number of the class A is less than
any number of the class B, the upper limit L of the numbers of the class A is at the
same time the lower limit of the numbers of the class B. It is clear, first of all, that
any number greater than L belongs to the class B. And if there were a number L'<L
belonging to the class B, then every number greater than L' would belong to the class B.
Hence every number less than L belongs to the class A, every number greater than L
belongs to the class B, and L itself may belong to either of the two classes.
t The word " closed " is used merely for emphasis. See § 2. — TRANS.
IV, §70J ALLIED GEOMETRICAL CONCEPTS 143
A = M — m is called the oscillation of the function f(x) in the
interval (a, b).
These definitions lead to several remarks. In order that a func
tion be finite in a closed interval (a, b~) it is not sufficient that it
should have a finite value for every value of x. Thus the function
defined in the closed interval (0, 1) as follows :
= 0, /(aj) = l/aj for x > 0,
has a finite value for each value of x ; biit nevertheless it is not
finite in the sense in which we have defined the word, for/(ce) > A
if we take x<l / A. Again, a function which is finite in the closed
interval (a, b) may take on values which differ as little as we please
from the upper limit M or from the lower limit m and still never
assume these values themselves. For instance, the function /(#),
defined in the closed interval (0, 1) by the relations
= 0, f(x) = l-x for 0<x<l,
has the upper limit M = 1, but never reaches that limit.
70. Properties of continuous functions. We shall now turn to the
study of continuous functions in particular.
THEOREM A. Letf(x) be a function which is continuous in the closed
interval (a, b) and e an arbitrary positive number. Then we can
always break up the interval (a, ft) into a certain number of partial
intervals in such a way that for any two values of the variable
whatever, x' and x", which belong to the same partial interval, we
always have \f(x') — f(x") \ < «•
Suppose that this were not true. Then let c=(a + ft)/2; at
least one of the intervals (a, c), (c, ft) would have the same prop
erty as (a, ft); that is, it would be impossible to break it up into
partial intervals which would satisfy the statement of the theorem.
Substituting it for the given interval (a, ft) and carrying out the
reasoning as above (§ 68), we could form an infinite sequence of
intervals (a, ft), (al} b^, (a2, ft2), • • -, each of which is half the preced
ing and has the same property as the original interval (a, ft). For
any value of n we could always find in the interval («n, ftn) two
numbers x' and x" such that |/(V)~/(X")I would be larger than e.
Now let X be the common limit of the two sequences of numbers
a, a1} a2> '" an^ b, b^ b2, • • •. Since the function /(#) is continuous
for x = X, we can find a number rj such that |/(.r) — /(X)| < e/2
144 DEFINITE INTEGRALS
[IV, § 70
whenever ja; — A| is less than rj. Let us choose n so large that both
an and bn differ from A by less than 77. Then the interval (an, bn}
will lie wholly within the interval (A - rj, A + 77) ; and if *' and a;"
are any two values whatever in the interval (an, Z>B), we must have
and hence |/<V) -/(.x") | < «. It follows that the hypothesis made
above leads to a contradiction ; hence the theorem is proved.
Corollary I. Let a, xlt x2, • • • , xp_1} b be a method of subdivision
of the interval (a, i) into p subintervals, which satisfies the con
ditions of the theorem. In the interval (a, a^) we shall have
!/(*) \ < I/O) I + « 5 and, in particular, \f(Xl) \ < |/(a) | + e. Like
wise, in the interval (xl} a-,) we shall have [/(*) I < j/(*i) + e,
and, a fortiori, \f(x) \ < |/(a) | + 2 c ; in particular, for x = xj,
|/(«0 I < I /(«) | + 2 e ; and so forth. For the last interval we shall
have
I/(*)|<|/(«P-I) | + €<|/(a) 1 + pe.
Hence the absolute value of f(x) in the interval (a, b~) always
remains less than |/(a) | + pe. It follows that every function which
is continuous in a closed interval (a, b) is finite in that interval.
Corollary II. Let us suppose the interval (a, b) split up into 7? sub-
intervals (a, x^, (xlt x2), ..., (xp_v b) such that \f(x') —f(x")\<e/2
for any two values of x which belong to the same closed subinterval.
Let 77 be a positive number less than any of the differences ^ — a,
x2 — Xi • • • , b — xp_l. Then let us take any two numbers whatever
in the interval (a, b) for which \x' - x" < ^ and let us try to find
an upper limit for (/(*') -/(*")). If the two numbers x' and x"
fall in the same subinterval, we shall have \f(x') —f(x")\< e/2.
If they do not, x' and x" must lie in two consecutive intervals,
and it is easy to see that |/<V) -f(x") \ < 2 (e/2) = c. Hence cor
responding to any positive number c another positive number rj can be
found such that
I/W-/CO.I<*
where x' and x" are any two numbers of the interval (a, b) for which
, [** — *"!< >?• This property is also expressed by saying that the
function /(x) is uniformly continuous in the interval (a, b).
THEOREM B. A function f(x) which is continuous in a closed
interval (a, b) takes on every value between /(a) and f(b) at least
once for some value of x which lies between a and b.
IV, §70] ALLIED GEOMETRICAL CONCEPTS 145
Let us first consider a particular case. Suppose that f(a) and
/(£) have opposite signs, — that /(a) < 0 and/(6) > 0, for instance.
We shall then show that there exists at least one value of x between
a and b for which f(x) = 0. Now/(x) is negative near a and posi
tive near b. Let us consider the assemblage of values of x between
a and b for which /(#) is positive, and let \ be the lower limit of
this assemblage (a < A < b*). By the very definition of a lower
limit /(A — A) is negative or zero for every positive value of h.
Hence /(A.), which is the limit of /(A — A), is also negative or zero.
But /(A) cannot be negative. For suppose that /(A) = — m, where
m is a positive number. Since the function /(x') is continuous for
x = A, a number rj can be found such that )/(#) —/(A) < m when
ever \x — A, | < rj, and the function f(x) would be negative for all
values of x between A. and A + rj. Hence A could not be the lower
limit of the values of x for which /(ic) is positive. Consequently
/(A) - 0.
Now let N be any number between /(a) and /(£>). Then the
function <£(#) =f(x) — N is continuous and has opposite signs for
x = a and x — b. Hence, by the particular case just treated, it
vanishes at least once in the interval (a, &).
THEOREM C. Every function which is continuous in a closed inter
val (a, b) actually assumes the value of its upper and of its lower
limit at least once.
In the first place, every continuous function, since we have
already proved that it is finite, has an upper limit M and a lower
limit m. Let us show, for instance, that f(x) — M for at least one
value of x in the interval (a, 5).
Taking c = (a + b)/2, the upper limit of f(x) is equal to M for
at least one of the intervals (a, e), (c, b). Let us replace (a, b)
by this new interval, repeat the process upon it, and so forth.
Reasoning as we have already done several times, we could form
an infinite sequence of intervals (a, b), (au &t), (o2, &2), • • • , each of
which is half the preceding and in each of which the upper limit of
f(x) is M. Then, if A is the common limit of the sequences a, aif
• • • , an, • • • and b, b1} • • • , bn, • • • , /(A) is equal to M. For suppose that
/(A) = M — h, where h is positive. We can find a positive number
rj such that f(x) remains between /(A) + h/2 and /(A) — h/ 2, and
therefore less than M — h/2 as long as x remains between A — rj
and A -f rj- Let us now choose n so great that an and bn differ from
their common limit A by less than 77. Then the interval (a,,, &„) lies
146 DEFINITE INTEGRALS [IV, § 71
wholly inside the interval (A. — 77, A. 4- *;), and it follows at once
that the upper limit of f(x) in the interval (an, bn) could not be
equal to M.
Combining this theorem with the preceding, we see that any func
tion which is continuous in a closed interval (a, ft) assumes, at least
once, every value between its upper and its lower limit. Moreover
theorem A may be stated as follows : Given a function which is
continuous in a closed interval (a, ft), it is possible to divide the inter
val into such small subreyions that the oscillation of the function in
any one of them will be less than an arbitrarily assigned positive
number. For the oscillation of a continuous function is equal to
the difference of the values of /(x) for two particular values of the
variable.
71. The sums S and s. Let /(#) be a finite function, continuous
or discontinuous, in the interval (a, ft), where a < b. Let us sup
pose the interval (a, b) divided into a number of smaller partial
intervals (a, o^), (a;u a;2), •••, (xp_l, b), where each of the numbers
xlt x2, • • •, #p_i is greater than the preceding. Let M and m be the
limits of f(x) in the original interval, and M{ and mi the limits
in the interval (a^-i, #,-), and let us set
S = M, (x, - a) + M2 (x2-x1)+--- + Mp (b - zp _t),
s = ml (xl — a) -f- 77*2(3-2 — xi)-\ H mP (* — XP-I)-
To every method of division of (a, b) into smaller intervals there
corresponds a sum S and a smaller sum s. It is evident that none
of the sums 5 are less than m(b — a), for none of the numbers Mi
are less than m ; hence these sums S have a lower limit /.* Like
wise, the sums s, none of which exceed M(b — a) have an upper
limit /'. We proceed to show that /' is at most equal to I. For this
purpose it is evidently sufficient to show that s^S' and s' 5j S, where
S, s and S', s' are the two sets of sums which correspond to any
two given methods of subdivision of the interval (a, b).
In the first place, let us suppose each of the subintervals (a, a^),
(#1, a2), • • • redivided into still smaller intervals by new points of
division and let
* If f(x) is a constant, S = s, M = m, and, in general, all the inequalities mentioned
become equations. — TRANS.
IV, §72] ALLIED GEOMETRICAL CONCEPTS 147
be the new suite thus obtained. This new method of subdivision
is called consecutive to the first. Let 2 and cr denote the sums anal
ogous to S and s with respect to this new method of division of the
interval (a, b), and let us compare S and s with 2 and a. Let us
compare, for example, the portions of the two sums 5 and 2 which
arise from the interval (a, a^). Let M[ and m[ be the limits of
f(x) in the interval (a, y^, M[ and m^ the limits in the interval
G/i> 1/2)9 "•» M'k an(i mk the limits in the interval (yt-i> %i)- Then
the portion of 2 which comes from (a, a^) is
and since the numbers M{, M%, ••-, M/. cannot exceed Mlt it is clear
that the above sum is at most equal to 3/j (xl — a). Likewise, the
portion of 2 which arises from the interval (xl} a:2) is at most equal
to M2 (x2 — «,), and so on. Adding all these inequalities, we find
that 2 = S, and it is easy to show in like manner that a- ^ s.
Let us now consider any two methods of subdivision whatever,
and let S, s and S', s' be the corresponding sums. Superimposing
the points of division of these two methods of subdivision, we get a
third method of subdivision, which may be considered as consecu
tive to either of the two given methods. Let 2 and <r be the sums
with respect to this auxiliary division. By the above we have the
relations
2<S, (r>s, 2<S', o->s';
and, since 2 is not less than a-, it follows that s'^ S and s^ S'. Since
none of the sums S are less than any of the sums s, the limit 7
cannot be less than the limit /' ; that is, / ^ /'.
72. Integrable functions. A function which is finite in an inter
val (a, b~) is said to be integrable in that interval if the two sums
S and 5 approach the same limit when the number of the partial
intervals is indefinitely increased in such a way that each of those
partial intervals approaches zero.
The necessary and sufficient condition that a function be integrable
in an interval is that corresponding to any positive number e another
number rj exists such that S — s is less than c whenever each of the
partial intervals is less than -r\.
This condition is, first, necessary, for if S and s have the same
limit 7, we can find a number ^ so small that | S — T\ and js — 7| are
148 DEFINITE INTEGRALS [iv, §72
each less than e/2 whenever each of the partial intervals is less
than 77. Then, a fortiori, S — s is less than e.
Moreover the condition is sufficient, for we may write *
s-s = s-i + i- r + r - s,
and since none of the numbers S — I, I — I', I' — s can be negative,
each of them must be less than e if their sum is to be less than e.
But since I — I' is a fixed number and e is an arbitrary positive
number, it follows that we must have /' = 7. Moreover S — I < e
and / — s < e whenever each of the partial intervals is less than 77,
which is equivalent to saying that S and s have the same limit 7.
The function /(#) is then said to be integrable in the interval
(a, £), and the limit 7 is called a definite integral. It is represented
by the symbol •
= ff(x)dx,
*J a
which suggests its origin, and which is read " the definite integral
from a to b of f(x) dx." By its very definition 7 always lies between
the two sums S and s for any method of subdivision whatever.
If any number between S and s be taken as an approximate value
of 7, the error never exceeds S — s.
Every continuous function is integrable.
The difference S — s is less than or equal to (b — a)«, where
w denotes the upper limit of the oscillation of f(x~) in the partial
intervals. But 77 may be so chosen that the oscillation is less than
a preassigned positive number in any interval less than 77 (§ 70).
If then 77 be so chosen that the oscillation is less than (./(b — a),
the difference S — s will be less than e.
Any monotonically increasing or monotonically decreasing function
in an interval is integrable in that interval.
A f unction /"( x) is said to increase monotonically in a given interval
(a, 6) if for any two values x', x" in that interval /(#') >/(«") when
ever x' > x". The function may be constant in certain portions of the
interval, but if it is not constant it must increase with x. Dividing
the interval (a, b) into n subintervals, each less than 77, we may write
S =/<X> (*! - a) +/(*2) (x2 _ aJt) + ••• +/(£) (b - *„_!),
S =/(a)(ar1 - a) +/(.T1)(*2 - xj -\- • • • +f(xn_,} (b - *„_,),
*For the proof that I and /' exist, see §73, which may be read before § 72. — TRANS.
IV, §72] ALLIED GEOMETRICAL CONCEPTS 149
for the upper limit of f(x) in the interval (a, a^), for instance,
is precisely f(x\), the lower limit /"(«); and so on for the other
subintervals. Hence, subtracting,
-s = (x1- a) [/(zj) -/(a)] + (*2 -
-/(*._,)].
None of the differences which occur in the right-hand side of this
equation are negative, and all of the differences xl — a, x2 — xl}
• • • are less than t] ; consequently
or
«-«<*[/(*) -/(«)],
and we need only take
in order to make 5 — <s < e. The reasoning is the same for a mono-
tonically decreasing function.
Let us return to the general case. In the definition of the inte
gral the sums S and s may be replaced by more general expres
sions. Given any method of subdivision of the interval (a, i) :
let £i* £s> ••*,£(>••• be values belonging to these intervals in order
(z, _ i = 4- = xi) • Then the sum
(2) .
evidently lies between the sums S and s, for we always have
7wf^/(£,.) 5 Af,-. If the function is integrable, this new sum has the
limit /. In particular, if we suppose that £1? £2, •••, ^n coincide
with a, ajj, •••, a;n_1, respectively, the sum (2) reduces to the sum
(1) considered above (§ 65).
There are several propositions which result immediately from the
definition of the integral. We have supposed that a < b ; if we now
interchange these two limits a and b, each of the factors xt — x{_!
changes sign; hence
Cf(x)dx = - Cf(x)dx.
Ja Jb
150 DEFINITE INTEGRALS [IV, §72
It also evidently follows from the definition that
f(x)dx = Cf(x)dx + f(x}dx,
f
•Jo.
at least if c lies between a and b; the same formula still holds when
I) lies between a and c, for instance, provided that the function f(x)
is integrable between a and c, for it may be written in the form
ff(x)dx = Cf(x)dx~ f f(x}dx = Cf(x)dx + fCf(x)dx.
Jo. Ja Jc J a Jb
If f(x) = A<f>(x) -f B\j/(x), where A and B are any two constants,
we have
/•> b /->h s*b
I f(x)dx = A I <f>(x)dx + B I ij/(x*)dx,
J a J a. \J a
and a similar formula holds for the sum of any number of functions.
The expression /(£,-) in (2) may be replaced by a still more gen
eral expression. The interval («-, I) being divided into n sub-
intervals (a, a^), • • • , (#/_!, a;,-), • • • , let us associate with each of the
subintervals a quantity £,, which approaches zero with the length
xt — xi_l of the subinterval in question. We shall say that £,
approaches zero uniformly if corresponding to every positive num
ber c another positive number -rj can be found independent of i and
such that j£,- < e whenever a\ — xf-1 is less than 77. We shall now
proceed to show that the sum
approaches the definite integral j^ffx^dx' as its limit provided
that £,- approaches zero uniformly. For suppose that rj is a number
so small that the two inequalities
are satisfied whenever each of the subintervals xi — xi_1 is less
than 17. Then we may write
/(*«-i)(«i-*«-i)- f
IV, §73] ALLIED GEOMETRICAL CONCEPTS 151
and it is clear that we shall have
< e + c(b — a)
S' - C f(x)dx
\J a
whenever each of the subintervals is less than 77. Thus the theorem
is proved.*
73. Darboux's theorem. Given any function f(x) which is finite in an inter
val (a, 6); the sums S and s approach their limits I and /', respectively, when
the number of subintervals increases indefinitely in such a way that each of
them approaches zero. Let us prove this for the sum S, for instance. We
shall suppose that a<6, and that/(x) is positive in the interval (a, 6), which can
be brought about by adding a suitable constant to/(x), which, in turn, amounts
to adding a constant to each of the sums S. Then, since the number / is the
lower limit of all the sums S, we can find a particular method of subdivision, say
a, zi, x2, • •-, Xp-i, 6,
for which the sum S is less than I + e/2, where e is a preassigned positive num
ber. Let us now consider a division of (a, 6) into intervals less than r;, and let us
try to find an upper limit of the corresponding sum S'. Taking first those inter
vals which do not include any of the points xlt x2, • • •, Xp_i, and recalling the
reasoning of § 71, it is clear that the portion of S' which comes from these inter
vals will be less than the original sum S, that is, less than I + e/2. On the other
hand, the number of intervals which include a point of the set Xi, x2, • • • , Xp_j
cannot exceed p — 1, and hence their contribution to the sum S' cannot exceed
(p — 1) Mil, where M is the upper limit of /(x). Hence
S'<I+e/2 + (p-l)Mr,,
and we need only choose r) less than e/2 M (p - 1) in order to make S'less than
I + f. Hence the lower limit I of all the sums /S is also the limit of any sequence
of <S's which corresponds to uniformly infinitesimal subintervals.
It may be shown in a similar manner that the sums s have the limit /'.
If the function /(x) is any function whatever, these two limits I and 7' are in
general different. In order that the function be integrable it is necessary and
sufficient that 7' = I.
74. First law of the mean for integrals. From now on we shall P
assume, unless something is explicitly said to the contrary, that
the functions under the integral sign are continuous.
* The above theorem can be extended without difficulty to double and triple inte
grals ; we shall make use of it in several places (§§ 80, 95, 97, 131, 144, etc.).
The proposition is essentially only an application of a theorem of Duhamel's
according to which the limit of a sum of infinitesimals remains unchanged when
each of the infinitesimals is replaced by another infinitesimal which differs from the
given infinitesimal by an infinitesimal of higher order. (See an article by W. F.
Osgood, Annals of Mathematics, 2d series, Vol. IV, pp. 161-178 : The Integral as
the Limit of a Sum and a Theorem of Duhamel's.)
152 DEFINITE INTEGRALS [IV, §74
Let f(x) and <£ (x) be two functions which are each continuous
in the interval (a, b), one of which, say <£(:*•), has the same sign
throughout the interval. And we shall suppose further, for the
sake of definiteness, that a < b and <f> (x) > 0.
Suppose the interval (a, b~) divided into subintervals, and let
£i> £2, • • • > £o • • • be values of x which belong to each of these
smaller intervals in order. All the quantities /(£,) lie between the
limits M and m of f(x) in the interval (a, b) :
Let us multiply each of these inequalities by the factors
respectively, which are all positive by hypothesis, and then add
them together. The sum S/(&) <£(&•) fa — zf_,) evidently lies
between the two sums ra 2 <£(£•) fa — o^) and 3/2 <£(£•) fa — £,._,).
Hence, as the number of subintervals increases indefinitely, we
have, in the limit,
m f $ (x} dx <, C /(» $ (x} dx < M C $ (x) dx,
Ja Jo. Ja
which may be written
Xb s*b
f(x}$(x}dx = p. \ $(x)dx,
J u.
where /x lies between m and M. Since the function f(x) is con
tinuous, it assumes the value /t for some value £ of the variable
which lies between a and b ; and hence we may write the preceding
equation in the form
(3) f /(*) <(> (x) dx = /(*) C $ (x) dx,
J a »/•
where £ lies between a and b.* If, in particular, <£(#) =1, the
integral JH* dx reduces to (b — a) by the very definition of an inte
gral, and the formula becomes
(4) f /(*)<& «(*-a)/tf).
* The lower sign holds in the preceding relations only when / (a-) = k. It is evident
that the formula still holds, however, and that a<£ < b in any case. —TRANS.
IV, §75] ALLIED GEOMETRICAL CONCEPTS 153
75. Second law of the mean for integrals. There is a second formula, due to
Bonnet, which he deduced from an important lemma of Abel's.
Lemma. Let e0, ei, • • • , ep be a set of monotonically decreasing positive quanti
ties, and MO , MI , • • • , Up the same number of arbitrary positive or negative quantities.
If A and B are respectively the greatest and the least of all of the sums s0 = u0,
s\ = UQ + «i , • • • i «p = w0 + MI + • • • + ut> , the sum
S — e0W0 + eiWi + • • • + epup
will lie between AeQ and -Be0) i-e. Aeo > S^ Bf0.
For we have
UQ — SO, Ui=Si — S0, •••, Up = Sp — Sp^i,
whence the sum S is equal to
So (*0 — «i) + Si (ei — e2) + • • • + Sp-i (ep-i - tp) + Spep.
Since none of the differences e0 — ei, ei — e2, • • •, fp~i — fp are negative, two
limits for S are given by replacing s0 , »i , • • • , sp by their upper limit A and then
by their lower limit B. In this way we find
S < A (e0 — ei + ei — e2 + • • • -f ep_i — ep + ev) = At0,
and it is likewise evident that S ^ Be0 .
Now let/(x) and <j> (x) be two continuous functions of x, one of which, 0 (a;),
is a positive monotonically decreasing function in the interval a < x < b. Then
the integral f^f(x) <j>(x)dx is the limit of the sum
/(a) <f> (a) (xi - a) + f(xi) <j> (xj) (x2 - xa) + . . . .
The numbers 0(a), 0(xi), • • • form a set of monotonically decreasing positive
numbers; hence the above sum, by the lemma, lies between A<f>(a) and B<f>(a),
where A and B are respectively the greatest and the least among the following
sums :
/(a) (X! - a) ,
/(a) (xi - a) +/(xi) (x2 - xi) ,
/(a) (X! - a) +/(xj) (x2 - Xi) + • • • +f(xn-l) (b - xn^).
Passing to the limit, it is clear that the integral in question must lie between
Ai<f,(a) and #10 (a), where AI and BI denote the maximum and the minimum,
respectively, of the integral f^f(x)dx, as c varies from a to b. Since this inte
gral is evidently a continuous function of its upper limit c (§ 76), we may write
the following formula :
(5) f /(x)0(x)cZx = 0(a) T
•J a Ja
When the function 0(x) is a monotonically decreasing function, without
being always positive, there exists a more general formula, due to Weierstrass.
In such a case let us set <f> (x) = <f> (b) + \fs (x). Then f (x) is a positive monoton
ically decreasing function. Applying the formula (5) to it, we find
f f(x)dx.
J a
154 DEFINITE INTEGRALS [IV, §76
From this it is easy to derive the formula
C f(x)+(x)dx = C f(x)+(b)dx + [0(o) - 0(6)] fV(x)
c/a J a Ja
f /(x) 0(z) dx = 0(a) f %s) <*z + 0(6) f bf(x) dx .
i/a "u ''
Similar formulae exist for the case when the function 0(x) is increasing.
76. Return to primitive functions. We are now in a position to
give a purely analytic proof of the fundamental existence theorem
(§ 67). Let/(x) be any continuous function. Then the definite integral
where the limit a is regarded as fixed, is a function of the upper
limit x. We proceed to show that the derivative of this function
isf(x). In the first place, we have
>x + A
f(t)dt,
= f
Jx
or, applying the first law of the mean (4),
where £ lies between x and x + h. As h approaches zero,
approaches /(x) ; hence the derivative of the function F(x) is /(x),
which was to be proved.
All other functions which have this same derivative are given
by adding an arbitrary constant C to F(x). There is one such
function, and only one, which assumes a preassigned value T/O for
x = a, namely, the function
When there is no reason to fear ambiguity the same letter x is
used to denote the upper limit and the variable of integration, and
/*/"(*) dx ig written in place of f*f(t) dt. But it is evident that
a definite integral depends only upon the limits of integration and
the form of the function under the sign of integration. The letter
which denotes the variable of integration is absolutely immaterial.
Every function whose derivative is /(x) is called an indefinite
integral of /(x), or & primitive function of /"(x), and is represented
by the symbol
r
f(x)dx,
IV, §70] ALLIED GEOMETRICAL CONCEPTS 155
the limits not being indicated. By the above we evidently have
Conversely, if a function F(x) whose derivative is /(x) can be
discovered by any method whatever, we may write
f(x)dx = F(aj)+ C.
In order to determine the constant C we need only note that the
left-hand side vanishes for x = a. Hence C = — F(a), and the
fundamental formula becomes
(6) f(x)dx = F(*)-F(a).
C/ U
If in this formula /(a- ) be replaced by F'(x), it becomes
F(a)-F(a)*= f F'(x)dx,
*J a
or, applying the first law of the mean for integrals,
where £ lies between a and x. This constitutes a new proof of the
law of the mean for derivatives ; but it is less general than the one
given in section 8, for it is assumed here that the derivative F'(a:) is
continuous.
We shall consider in the next chapter the simpler classes of func
tions whose primitives are known. Just now we will merely state
a few of those which are apparent at once :
A(x — a)adx = A - — — (- (', a-fl^O;
dx
I cos x dx = sin x -f C ; I sin x dx = — cos x -f- C ;
xdx = h C, m =£ 0;
150 DEFINITE INTEGRALS [IV, §76
= log/(x) + C.
The proof of the fundamental formula (6) was based upon the
assumption that the function f(x) was continuous in the closed inter
val (a, b). If this condition be disregarded, results may be obtained
which are paradoxical. Taking f(x) == I/a;2, for instance, the for
mula (6) gives
f"^ = i_i.
} x2 a I
\J a
The left-hand side of this equality has no meaning in our present
system unless a and b have the same sign ; but the right-hand side
has a perfectly determinate value, even when a and b have different
signs. We shall find the explanation of this paradox later in the
study of definite integrals taken between imaginary limits.
Similarly, the formula (6) leads to the equation
r
1
^
/(*> °/(«)
If /(a) and/(£) have opposite signs, f(x) vanishes between a and b,
and neither side of the above equality has any meaning for us at
present. We shall find later the signification which it is convenient
to give them.
Again, the formula (6) may lead to ambiguity. Thus, if
/(») = !/(!+ *2), we find
= arc tan b — arc tan a.
Here the left-hand side is perfectly determinate, while the right-
hand side has an infinite number of determinations. To avoid this
ambiguity, let us consider the function
This function F(x) is continuous in the whole interval and van
ishes with x. Let us denote by arc tan x, on the other hand, an
angle between - Tr/2 and + Tr/2. These two functions have the
iv, §77] ALLIED GEOMETRICAL CONCEPTS 157
same derivative and they both vanish for x = 0. It follows that
they are equal, and we may write the equality
rb dx rb dx ra dx
\ : = I ; — | -. = arc tan b — arc tan a,
Jm 1 + x* J0 1 + x* J0 1 + x*
where the value to be assigned the arctangent always lies between
-7T/2 and +7T/2.
In a similar manner we may derive the formula
>b dx
f
I/O
= arc sin b — arc sin a,
where the radical is to be taken positive, where a and b each lie
between — 1 and + 1, and where arc sin x denotes an angle which
lies between — Tr/2 and + Tr/2.
77. Indices. In general, when the primitive F(x) is multiply determinate, we
should choose one of the initial values F(a) and follow the continuous variation
of this branch as x varies from a to b. Let us consider, for instance, the integral
fobzar*. f-^w-
J. ^+<? J. i +/•'(*)
where
and where P and Q are two functions which are both continuous in the interval
(a, b) and which do not both vanish at the same time. If Q does not vanish
between a and 6, /(x) does not become infinite, and arc tan/(x) remains between
— 7f/2 and + rt/2. But this is no longer true, in general, if the equation Q = 0
has roots in this interval. In order to see how the formula must be modified, let
us retain the convention that arc tan signifies an angle between — if/2 and + if/2,
and let us suppose, in the first place, that Q vanishes just once between a and b
for a value x = c. We may write the integral in the form
b
r
Ja
f'(x)dx
where e and e' are two very small positive numbers. Since /(x) does not become
infinite between a and c — e, nor between c + e' and 6, this may again be written
f'dx
• = arc tan/(c — e) — - arc tan/(a)
fC+t
+ arc tan/ (6) - arc tan/(c + e') + I
Jc-e
Several cases may now present themselves. Suppose, for the sake of definite-
ness, that/(x) becomes infinite by passing from + oo to — oo. Then/(c — e) will
be positive and very large, and arc tan /(c — e) will be very near to ir/2; while
158 DEFINITE INTEGRALS [IV, § 78
/(c + e') will be negative and very large, and arc tan/(c + e7) will be very near
— 7T/2. Also, the integral J^lV wil1 be verv small in absolute value; and,
passing to the limit, we obtain the formula
f
f(x)dx
= TT + arc tan/(6) — arctan/(a).
Similarly, it is easy to show that it would be necessary to subtract n if /(x)
passed from — co to + ». In the general case we would divide the interval
(a, 6) into subintervals in such a way that /(x) would become infinite just once
in each of them. Treating each of these subintervals in the above manner and
adding the results obtained, we should find the formula
£
*s a
f'(x) dx
. ' 'T = arc tan/(6) - arc tan/(o) + (K - K') x,
i + j (x)
where K denotes the number of times that /(x) becomes infinite by passing from
+ oo to — co, and K' the number of times that /(x) passes from — oo to + oo.
The number K — K' is called the index of the function /(x) between a and 6.
When/(x) reduces to a rational function Vi/V, this index may be calculated
by elementary processes without knowing the roots of V. It is clear that we
may suppose Vi prime to and of less degree than V, for the removal of a poly
nomial does not affect the index. Let us then consider the series of divisions
necessary to determine the greatest common divisor of Fand FI, the sign of the
remainder being changed each time. First, we would divide V by FI, obtaining
a quotient Qi and a remainder — F2. Then we would divide FI by F2, obtaining a
quotient Q2 and a remainder — Vs ; and so on. Finally we should obtain a con
stant remainder — F» + 1. These operations give the following set of equations :
F = FiQi - F2,
F! = F2Q2 - F8,
The sequence of polynomials
(7) F, FI, Fa, ••-, Vr.it rr, Fr + 1, -.., Fn, Fn + 1
has the essential characteristics of a Sturm sequence : 1) two consecutive poly
nomials of the sequence cannot vanish simultaneously, for if they did, it could
be shown successively that this value of x would cause all the other polynomials
to vanish, in particular Vn + \; 2) when one of the intermediate polynomials FI,
Pai •••,Vn vanishes, the number of changes of sign in the series (7) is not altered,
for if Fr vanishes for x = c, Fr_i and Vr + \ have different signs for x — c. It
follows that the number of changes of sign in the series (7) remains the same,
except when x passes through a root of F = 0. If Fi/F passes from + oo to — oo,
this number increases by one, but it diminishes by one on the other hand if
V\/V passes from — co to +00. Hence the index is equal to the difference of
the number of changes of sign in the series (7) for x = 6 and x = a.
78. Area of a curve. We can now give a purely analytic definition
of the area bounded by a continuous plane curve, the area of the
rectangle only being considered known. For this purpose we need
IV, §78] ALLIED GEOMETRICAL CONCEPTS 159
only translate into geometrical language the results of § 72. Let
f(x) be a function which is continuous in the closed interval (a, b),
and let us suppose for definiteness that a < b and that f(x) > 0 in
the interval. Let us consider, as above (Fig. 9, § 65), the portion of
the plane bounded by the contour AMBB0A0, composed of the seg
ment A0B0 of the x axis, the straight lines AA0 and BB0 parallel to
the y axis, and having the abscissae a and b, and the arc of the curve
A MB whose equation is y =f(x). Let us mark off on A0BQ a certain
number of points of division Pl} P2, • • • , P,_i, Pi} •••, whose abscissae
are x1} x2, • • •, #;_!, xi} •••, and through these points let us draw
parallels to the y axis which meet the arc A MB in the points
Qi, #2> •••, Qi-D 0>i> •••) respectively. Let us then consider, in
particular, the portion of the plane bounded by the contour
Qi-iQiPiPi-iQi-i, an(i let us mark upon the arc Q.-^Q,- the highest
and the lowest points, that is, the points which correspond to the
maximum M{ and to the minimum m{ of f(x) in the interval
(#,•_!, a;,-). (In the figure the lowest point coincides with <2,-_j.)
Let Rf be the area of the rectangle Pi-lPisisi_1 erected upon the
base Pi-i-P, with the altitude JJ/f, and let r{ be the area of the
rectangle Pf-iP^Q,--! erected upon the base P,-^^ with the alti
tude mt. Then we have
and the results found above (§ 72) may now be stated as follows :
whatever be the points of division, there exists a fixed number /
which is always less than 2A\ and greater than 2r,., and the two
sums 2Ri and 2rf approach / as the number of sabintervals Pi^1Pi
increases in such away that each of them approaches zero. We shall
call this common limit I of the two sums 2-Ri and 2r{ the area of
the portion of the plane bounded by the contour AMBB0A0A. Thus
the area under consideration is defined to be equal to the definite
integral I j (<£) ax,
This definition agrees with the ordinary notion of the area of a
plane curve. For one of the clearest points of this rather vague
notion is that the area bounded by the contour Pi-iPiQiniQi_]Pi_l
lies between the two areas Rf and r( of the two rectangles Pj_iP,-*,-s»-i
and Pi-iPiiiQi-i; hence the total area bounded by the contour
AMBB0A0A must surely be a quantity which lies between the two
sums 2/?f and 2r,-. But the definite integral / is the only fixed quan
tity which always lies between these two sums for any mode of
subdivision of AQB0, since it is the common limit of 2-R, and 2r<.
160
DEFINITE INTEGRALS
[IV, § 79
where & is any value whatever in the interval (xt_1} a;,).
element
The given area may also be defined in an infinite number of other
ways as the limit of a sum of rectangles. Thus we have seen that
the definite integral / is also the limit of the sum
But the
of this sum represents the area of a rectangle whose base is P.-.jPj
and whose altitude is the ordinate of any point of the arc Qi_lniQi.
It should be noticed also that the definite integral / represents
the area, whatever be the position of the arc AMR with respect to
the x axis, provided that we adopt the convention made in § 67.
Every definite integral therefore represents an area ; hence the calcu
lation of such an integral is called a quadrature.
The notion of area thus having been made rigorous once for all,
there remains no reason why it should not be used in certain
arguments which it renders nearly intuitive. For instance, it is
perfectly clear that the area considered above lies between the areas
of the two rectangles which have the common base A0B0, and which
have the least and the greatest of the ordinates of the arc A MB,
respectively, as their altitudes. It is therefore equal to the area of
a rectangle whose base is A0B0 and whose altitude is the ordinate
of a properly chosen point upon the arc AMB, — which is a restate
ment of the first law of the mean for integrals.
79. The following remark is also important. Let f(x) be a func
tion which is finite in the interval (a, b) and which is discontinuous
in the manner described below for
a finite number of values between
a and b. Let us suppose that /(a-)
is continuous from c to c + &(&>0),
and that f(c -f c) approaches a cer
tain limit, which we shall denote
f(c + 0), as e approaches zero
through positive values ; and like
wise let us suppose that f(x) is
continuous between c — k and c and that/(c - c) approaches a limit
f(c - 0) as e approaches zero through positive values. If the two
limits f(c + 0) and f(c - 0) are different, the function f(x) is dis
continuous for x = c. It is usually agreed to take for /(c) the
n
FIG. 11
IV, §80] ALLIED GEOMETRICAL CONCEPTS 161
value [f(c 4- 0) +f(c — 0)]/2. If the function /(a;) has a certain
number of points of discontinuity of this kind, it will be repre
sented graphically by several distinct arcs AC, C'D, D'B. Let c
and d, for example, be the abscissae of the points of discontinuity.
Then we shall write
Xb /~> c s*d s*b
f(x)dx = I f(x)dx + I f(x)dx + I f(x)dx,
i/a *J c i/a*
in accordance with the definitions of § 72. Geometrically, this definite
integral represents the area bounded by the contour A CC'DD'BB0A0A.
If the upper limit b now be replaced by the variable x, the definite
integral
is still a continuous function of x. In a point x where f(x) is con
tinuous we still have F'(x')=f(x'). For a point of discontinuity,
x = c for example, we shall have
S>C+fl
F(c + K) - F(c) = I f(x) dx = hf(c + BK), 0 < $ < 1,
and the ratio \_F(c + A) — F(c)]/h approaches f(c + 0) or f(c — 0)
according as h is positive or negative. This is an example of a
function F(x) whose derivative has two distinct values for certain
values of the variable.
80. Length of a curvilinear arc. Given a curvilinear arc AB; let us
take a certain number of intermediate points on this arc, m1, m2)
•••, mn-\, and let us construct the broken line A?n1m2 • • • mn_lB by
connecting each pair of consecutive points by a straight line.
If the length of the perimeter of this broken line approaches a
limit as the number of sides increases in such a way that each of
them approaches zero, this limit is defined to be the length of the
arc AB.
Let
be the rectangular coordinates of a point of the arc AB expressed
in terms of a parameter t, and let us suppose that as t varies from
a to b (a < b') the functions /, <£, and \j/ are continuous and possess
continuous first derivatives, and that the point (x, y, z) describes
the arc AB without changing the sense of its motion. Let
DEFINITE INTEGRALS [IV, §80
be the values of t which correspond to the vertices of the broken
line. Then the side ct is given by the formula
or, applying the law of the mean to x{ — #,_!, • • -,
where £,., 77,, & lie between £,._! and tt. When the interval (£,._!, /,)
is very small the radical differs very little from the expression
In order to estimate the error we may write it in the form
A'.-- .)] + • • •
But we have
!/'(£) + 1 /'(**- 0 !
and consequently
Hence, if each of the intervals be made so small that the oscillation
of each of the functions /'(*), <f>'(t), ^'(*) is less than c/3 in any
interval, we shall have
where
M<«J
and the perimeter of the broken line is therefore equal to
The supplementary term 2e,-(£,- — £,-_,) is less in absolute value
than e2(fc — #,- ._,), that is, than c(/> — a). Since e may be taken as
small as we please, provided that the intervals be taken sufficiently
small, it follows that this term approaches zero ; hence the length S
of the arc AB is equal to the definite integral
(8) s=C
c/ (l
This definition may be extended to the case where the derivatives
/', <f>', $ are discontinuous in a finite number of points of the arc AB,
IV, §80] ALLIED GEOMETRICAL CONCEPTS 163
which occurs when the curve has one or more corners. We need only
divide the arc AB into several parts for each of which/', <£', ^' are
continuous.
It results from the formula (8) that the length S of the arc
between a fixed point A and a variable point M, which corresponds
to a value t of the parameter, is a function of t whose derivative is
whence, squaring and multiplying by dt2, we find the formula
(9) dS2 = dx2 + dy* + dz2,
which does not involve the independent variable. It is also easily
remembered from its geometrical meaning, for it means that dS is
the diagonal of a rectangular parallelepiped whose adjacent edges are
dx, dy, dz.
Note. Applying the first law of the mean for integrals to the
definite integral which represents the arc M0M1} whose extremities
correspond to the values t0, ^ of the parameter (^ > *0), we find
s = arc JUoJ/i = (t, - t0) V/'2(0) + 4,'2(0) + ,/,"(0),
where 0 lies in the interval (t0, ^). On the other hand, denoting
the chord MnM^ by c, we have
c2 = [/CO -/Co)]2
Applying the law of the mean for derivatives to each of the differ
ences f(ti)—f(tn), •••, we obtain the formula
where the three numbers £, rj, £ belong to the interval (£0, ^). By
the above calculation the difference of the two radicals is less than e,
provided that the oscillation of each of the functions/'^), <£'(£),
is less than e/3 in the interval (#0, ^). Consequently we have
or, finally,
1-
s
If the arc MQMl is infinitesimal, tl— t0 approaches zero; hence c,
and therefore also 1— c/s, approaches zero. It follows that the ratio
of an infinitesimal arc to its chord approaches unity as its limit.
164 DEFINITE INTEGRALS [IV, §81
Example. Let us find the length of an arc of a plane curve whose
equation in polar coordinates is p = /(w). Taking <o as independent
variable, the curve is represented by the three equations x = p cos w,
y = p sin o>, z = 0 ; hence
ds2 = dx2 + dy2 = (cos to dp — p sin o> e?a>)2 + (sin wdp -}- p cos to c?w)2,
or, simplifying,
ds2 = dp2 + P2d^2.
Let us consider, for instance, the cardioid, whose equation is
p = R + R cos a).
By the preceding formula we have
ds2 = R2 dta2 [sin2 w + (1 + cos o>)2] = 4 R2 cos2 ^ do2,
or, letting o> vary from 0 to TT only,
ds = 2 R cos — f/o) ;
U
and the length of the arc is
f
&R sin
where w0 and u^ are the polar angles which correspond to the extrem
ities of the arc. The total length of the curve is therefore 8 R.
81. Direction cosines. In studying the properties of a curve we are
often led to take the arc itself as the independent variable. Let us
choose a certain sense along the curve as positive, and denote by s
the length of the arc AM between a certain fixed point A and a vari
able point M, the sign being taken + or — according as M lies in
the positive or in the negative direction from A. At any point M
of the curve let us take the direction of the tangent which coincides
with the direction in which the arc is increasing, and let a, ft, y be
the angles which this direction makes with the positive directions
of the three rectangular axes Ox, Oy, OK. Then we shall have the
following relations :
COS a _ COS ft _ COS y 1 ±1
dx dy dz - Vrfz2 + dif + dz2 ~ ds '
To find which sign to take, suppose that the positive direction of
the tangent makes an acute angle with the x axis ; then x and s
increase simultaneously, and the sign + should be taken. If the
angle a is obtuse, cos a is negative, x decreases as s increases, dx/ds
IV, §82] ALLIED GEOMETRICAL CONCEPTS 165
is negative, and the sign -f- should be taken again. Hence in any
case the following formulae hold :
dx dy dz
(10) cos a = — > cos B = ~r » cos y = — - >
as as ds
where dx, dy, dz, ds are differentials taken with respect to the same
independent variable, which is otherwise arbitrary.
82. Variation of a segment of a straight line. Let MM^ be a segment
of a straight line whose extremities describe two curves C, Cx. On
each of the two curves let us choose a
point as origin and a positive sense of
motion, and let us adopt the follow
ing notation : s, the arc AM; s1} the arc
AlMl, — the two arcs being taken with
the same sign ; I, the length M Ml ; B, the
angle between M M^ and the positive di
rection of the tangent MT; 01} the angle
between AT, M and the positive direction
FIG 12
of the tangent M^ 7\. We proceed to
try to find a relation between $, 61 and the differentials ds, dsl} dl.
Let (x, y, z), (x1} ylt z^ be the coordinates of the points M, Mly
respectively, a, ft, y the direction angles of MT, and a^, fa, yl the
direction angles of M^ 7\. Then we have
P = (x - arO" + (y - y^ + (z - ztf,
from which we may derive the formula
ldl = (x- x^ (dx — dxj + (y - y,) (dy - dyj + (z - z^ (dz — dzj,
which, by means of the formulae (10) and the analogous formulae
for Clf may be written in the form
_
ds
Ix — x, 11 — ?/, z — z, \
dl = I - — - C03 a + J Jl COS (3 H -- j—1 COS y I
\ V C l> /
/x, — x •?/, — 11 z, — z
+ I -L^ — cos <*! -f >/i cos ft + -*-j — cos
\ 6 (/ l>
But (a; — x^)/l, (y — y\) fl, (z — z-^/l are the direction cosines of
M± M, and consequently the coefficient of ds is — cos 0. Likewise
the coefficient of dsl is — cos ^; hence the desired relation is
(10') dl = — ds cos 6 — ds! cos 0,.
We shall make frequent applications of this formula ; one such we
proceed to discuss immediately.
166 DEFINITE INTEGRALS [IV, §83
83. Theorems of Graves and of Chasles. Let E and E' be two confocal ellipses,
and let the two tangents MA, MB to the interior ellipse E be drawn from a point
M, which lies on the exterior ellipse E'. The
difference MA + MB — arc ANB remains con
stant as the point M describes the ellipse E'.
Let « and s' denote the arcs OA and OB,
<r the arc O'M, I and I' the distances AM and
BM, 6 the angle between MB and the positive
direction of the tangent M T. Since the ellipses
are confocal the angle between MA and M T is
FlQ 13 equal to it — 6. Noting that AM coincides
with the positive direction of the tangent at A,
and that BM is the negative direction of the tangent at B, we find from the
formula (10'), successively,
dl = — ds + d<r cos 6 ,
dl' — ds' — do- cos 6 •
whence, adding,
d(l + l')=d (s1 -s)=d (arc ANB),
which proves the proposition stated above.
The above theorem is due to an English geometrician, Graves. The following
theorem, discovered by Chasles, may be proved in a similar manner. Given an
ellipse and a confocal hyperbola which meets it at N. If from a point M on that
branch of the hyperbola which passes through N the two tangents MA and MB
be drawn to the ellipse, the difference of the arcs NA — NB will be equal to the
difference of the tangents MA — MB.
III. CHANGE OF VARIABLE INTEGRATION BY PARTS
A large number of definite integrals which cannot be evaluated
directly yield to the two general processes which we shall discuss
in this section.
84. Change of variable. If in the definite integral /*/(*) dx the
variable x be replaced by a new independent variable t by means
of the substitution x = <f>(t), a new definite integral is obtained.
Let us suppose that the function <f>(t) is continuous and possesses a
continuous derivative between a and ft, and that <f>(f) proceeds from
a to b without changing sense as t goes from a to ft.
The interval (a, ft) having been broken up into subintervals by
the intermediate values a, tv, t», • • •, tn_l, ft, let a, xl} xz, • • •, xn_l} b
be the corresponding values of x == <f>(t). Then, by the law of the
mean, we shall have
where Bt lies between ti_l and ?,-. Let £,- — <£(0,-) be the corresponding
value of x which lies between xi_l and x(. Then the sum
IV. 5*»1 CHANGE OF VARIABLE 167
(x, - a) + /(&) (x, - *t) + • • • +/(£,) (6 - xn _ 0
approaches the given definite integral as its limit. But this sum
may also be written
and in this form we see that it approaches the new definite integral
C'
Ja
as its limit. This establishes the equality
Cb
(\Y\ I f(x~\dx —
\-L-LJ t ,/ v v "
v a ^
which is called the formula for the change of variable. It is to
be observed that the new differential under the sign of integration
is obtained by replacing x and dx in the differential f(x}dx by their
values <f>(t) and <}>'(t)dt, while the new limits of integration are the
values of t which correspond to the old limits. By a suitable choice
of the function <£(£) the new integral may turn out to be easier to
evaluate than the old, but it is impossible to lay down any definite
rules in the matter.
Let us take the definite integral
/
Jo
dx
(x - a)* + p*
for instance, and let us make the substitution x = a -f- fit. It
becomes
dx 1 r ' dt I a
tan * + arc tan
or, returning to the variable x,
1 / x — a a
-;l arc tan — - -- \- arc tan -
Xot all the hypotheses made in establishing the formula (11) were
necessary. Thus it is not necessary that the function <£(£) should
always move in the same sense as t varies from a to f3. For defi-
niteness let us suppose that as t increases from a to y (y < /8), <£(£)
steadily increases from a to c (c > i) ; then as t increases from y to
/3, <£(£) decreases from c to I. If the f unction /(x) is continuous in
the interval (a, c), the formula may be applied to each of the inter
vals (a, c), (c, b), which gives
168 DEFINITE INTEGRALS L*v,$84
or, adding,
On the other hand, it is quite necessary that the function <f>(t)
should be uniquely denned for all values of t. If this condition be
disregarded, fallacies may arise. For instance, if the formula be
applied to the integral f_l dx, using the transformation x = 1?/2,
we should be led to write
/ + i rl 3
*-J j
which is evidently incorrect, since the second integral vanishes. In
order to apply the formula correctly we must divide the interval
(— 1, -f- 1) into the two intervals (— 1, 0), (0, 1). In the first of
these we should take x = — Vr and let t vary from 1 to 0. In the
second half interval we should take x = ~\/ts and let t vary from
0 to 1. We then find a correct result, namely
X + i <~i
dx = 3 I
1 t/O
Note. If the upper limits b and ft be replaced by x and t in the
formula (11), it becomes
which shows that the transformation x = <£(£) carries a function
F(x), whose derivative is /(#), into a function <£(£) whose derivative
is /[<£(£)]<£'(£). This also follows at once from the formula for the
derivative of a function of a function. Hence we may write, in
general,
which is the formula for the change of variable in indefinite
integrals.
ivr, §85] INTEGRATION BY PARTS 169
85. Integration by parts. Let u and v be two functions which,
together with their derivatives u' and v', are continuous between a
and b. Then we have
d(uv) _ dv du
dx dx dx
whence, integrating both sides of this equation, we find
C dhm) Cb dv Cb du
\ dx = I u — dx+ \ v ----- dx.
Ja dx Ja dx Ja dx
This may be written in the form
f*b /*b
(12) / u dv = \_uv~\l - I v du,
\J a *J a
where the symbol [F(x)]« denotes, in general, the difference
If we replace the limit b by a variable limit x, but keep the limit a
constant, which amounts to passing from definite to indefinite inte
grals, this formula becomes
(13) | u dv = uv — I v du.
Thus the calculation of the integral / u dv is reduced to the cal
culation of the integral fvdu, which may be easier. Let us try,
for example, to calculate the definite integral
r
I xmlogxdx, ra + 1^0.
\J ct
Setting u = logic, v — xm + l/(m + 1), the formula (12) gives
c\ n^+'iogarr i rb
logx.xmdx=\ — I xmdx
Ja L m + 1 J « in + 1 Ja
"1 + 1 log X Xm + * 1 6
_
m+1 ~(m+l)2a'
This formula is not applicable if m + 1 = 0 ; in that particular
case we have
It is possible to generalize the formula (12). Let the succes
sive derivatives of the two functions u and v be represented by
u', M", -.., w(" + 1>; v', v", ••-, v(n + 1\ Then the application of the
170 DEFINITE INTEGRALS [IV, §85
formula (12) to the integrals fudv™, fu'dv<*-", ••• leads to the
following equations :
/>6 s*b
I uv(n + 1)dx= I udv^ = [>w(n)]*
Ja Ja
s*\> r>t> fb
I u'v^dx =1 M'^(--« = |>'r<" -''I*- I wV
Ja Ja J*
~b /-.b
/ u<*>v'dx =1 u^do =[«('°"]a
Ja Ja.
Multiplying these equations through by + 1 and — 1 alternately,
and then adding, we find the formula
» + l C (n
Ja
which reduces the calculation of the integral fuv^n+l)dx to the cal
culation of the integral f*i<H + l)vdx.
In particular this formula applies when the function under the
integral sign is the product of a polynomial of at most the wth
degree and the derivative of order (n + 1) of a known function v.
For then w(M + 1) = 0, and the second member contains no integral
signs. Suppose, for instance, that we wished to evaluate the definite
integral
fW(*)*«,
\J a
where /(x) is a polynomial of degree n. Setting u =/(z), v = ewi/u)n+v,
the formula (14) takes the following form after e"x has been taken
out as a factor :
The same method, or, what amounts to the same thing, a series of
integrations by parts, enables us to evaluate the definite integrals
I c,osmxf(x)dx, I sinmxf(x)dx,
J a */m
where f(x) is a polynomial.
IV, §86] INTEGRATION BY PARTS 171
86. Taylor's series with a remainder. In the formula (14) let us
replace u by a function F(x) which, together with its first n + 1
derivatives, is continuous between a and b, and let us set v = (b — x)n.
Then we have
v' = — n(b — a;)"-1, v" = n(n - l)(b -re)"-2, •-.,
v<»> = (-!)»!. 2 .--W, y(" + 1> = 0,
and, noticing that v, v', v", •••, i/"-1) vanish for a: = b, we obtain the
following equation from the general formula :
0 = (-!)» n\F(b) — n\F(a) -n\F'(a)(b -a)
ri\ ~]
-jF't (a) (b -a)2 ---- F<«> (a) (b - a)»
» ]
which leads to the equation
n, — ay
- - ^
i rb
-7 / ^'i + I)(a-)(&-a;)n^.
?i!i/a
Since the factor (i — x)n keeps the same sign as x varies from a to
b, we may apply the law of the mean to the integral on the right,
which gives
I F« + l\x)(b - x)»dx = F<» + 1'(f) f (b- x}ndx
Ja Ja
where £ lies between a and b. Substituting this value in the preced
ing equation, we find again exactly Taylor's formula, with Lagrange's
form of the remainder.
87. Transcendental character of e . From the formula (15) we can prove a
famous theorem due to Hermite : The number e is not a root of any algebraic
equation whose coefficients are all integers.*
Setting a = 0 and w = — 1 in the formula (15), it becomes
JT «-»
* The present proof is due to D. Hilbert, who drew his inspiration from the method
used by Hermite.
172 DEFINITE INTEGRALS [IV, §87
where
F(x) =/(x) +/'(z)
and this again may be written in the form
(16) F(b) =
Now let us suppose that e were the root of an algebraic equation whose coeffi
cients are all integers :
c0 + c\e + c2e2 + ---- 1- cmem = 0.
Then, setting b = 0, 1, 2, • • • , wi, successively, in the formula (16), and adding
the results obtained, after multiplying them respectively by c0, cl5 • • •, cm, we
obtain the equation
(17)
e~*dx =
where the index i takes on only the integral values 0, 1, 2, • • • , m. We proceed
to show that such a relation is impossible if the polynomial /(x), which is up to
the present arbitrary, be properly chosen.
Let us choose it as follows :
/(X) - - 1 - XP~I(X - l)p(x-2)P---(x -m)P,
(P - I)1-
where p is a prime number greater than m. This polynomial is of degree
mp -|- p — i? and all of the coefficients of its successive derivatives past the pth
are integral multiples of p, since the product of p successive integers is divisible
by p!. Moreover /(x), together with its first (p - 1) derivatives, vanishes for
x = 1, 2, • • -, m, and it follows that F(l), F(2), • ••, F(m) are all integral mul
tiples of p. It only remains to calculate F(0), that is,
=/(0)
In the first place, /(O) =/(0) = • • • = /O- 2>(0) = 0, while /Cp>(0), /^ + 1>(0), • • •
are all integral multiples of p, as we have just shown. To find /</' - 1) (0) we need
only multiply the coefficient of XP~I in/(x) by (p - 1) !, which gives ± (1 . 2 • • • m)p.
Hence the sum
c0F(0)
is equal to an integral multiple of p increased by
i- c0(l . 2 • • • m)p.
If p be taken greater than either m or c0, the above number cannot be divisible
by p ; hence the first portion of the sum (17) will be an integer different from zero.
We shall now show that the sum
can be made smaller than any preassigned quantity by taking p sufficiently large.
As x varies from 0 to i each factor of /(x) is less than m ; hence we have
IV, §88] INTEGRATION BY PARTS 173
u
f(x)e-*dx
. mmp+P-\ | e~xdx<
(p-1)! Jo (P-1)I
from which it follows that
2««<f/(x)<
«/o
where M is an upper limit of | c0 + | Ci | + • • • + | cm | . As p increases indefi
nitely the function 0(p) approaches zero, for it is the general term of a conver
gent series in which the ratio of one term to the preceding approaches zero. It
follows that we can find a prime number p so large that the equation (17) is
impossible ; hence Hermite's theorem is proved.
88. Legendre's polynomials. Let us consider the integral
where Pn(x) is a polynomial of degree n and Q is a polynomial of degree less
than n, and let us try to determine Pn(x) in such a way that the integral van
ishes for any polynomial Q. We may consider Pn (x) as the nth derivative of a
polynomial R of degree 2n, and this polynomial R is not completely determined,
for we may add to it an arbitrary polynomial of degree (n — 1) without changing
its nth derivative. We may therefore set Pn= dnR/dxn, where the polynomial E,
together with its first (n — 1) derivatives, vanishes for x = a. But integrating
by parts we find
rQdnEdx-
Ja Q^» -
and since, by hypothesis,
E(o)=0, B'(a) = 0, ••-, B(»-J)(o)=0,
the expression
Q (6) R(» - 1) (6) - Q'(b) fi(» - 2) (6) + • • • ± Qf.* ~ 0 (b) R (b)
must also vanish if the integral is to vanish.
Since the polynomial Q of degree n — 1 is to be arbitrary, the quantities
Q(&)i Q'(&)> • ••> Qf-n~l)(b) are themselves arbitrary; hence we must also have
B(6) = 0, R'(b) = 0, ••-, E<«-i)(6) = 0.
The polynomial R (x) is therefore equal, save for a constant factor, to the product
(x - a)n (x - b)n ; and the required polynomial Pn (x) is completely determined,
save for a constant factor, in the form
If the limits a and 6 are — 1 and + 1, respectively, the polynomials Pn are
Legendre's polynomials. Choosing the constant C with Legendre, we will set
(18) Xn = - - - — [(x2 - !)»].
2.4.6...2nax»LV
174 DEFINITE INTEGRALS [IV, §88
If we also agree to set X0 — 1, we shall have
y i y- r 3x*~l r 5x3 -3x
-AO = 1, -<M = x, JLZ = — » Xs = > ••••
"2i 2i
In general, Xn is a polynomial of degree n, all the exponents of x being even or
odd with n. Leibniz' formula for the nth derivative of a product of two factors
(§ 17) gives at once the formulae
(19) -Z»(l) = l, -T.(-l) = (-!)"•
By the general property established above,
(20) C + X,t<t>(x)dx = 0,
«/ — i
where tf> (x) is any polynomial of degree less than n. In particular, if m and n
are two different integers, we shall always have
(21) C +
J-i
This formula enables us to establish a very simple recurrent formula between
three successive polynomials Xn. Observing that any polynomial of degree n
can be written as a linear function of X0, Xi, • • • , Xn, it is clear that we may set
where C0, Ci, C2, • • • are constants. In order to find C3, for example, let us
multiply both sides of this equation by ^Tn_2, and then integrate between the
limits — 1 and + 1. By virtue of (20) and (21), all that remains is
C + ' 2
3J_1 "" "~2
and hence C3 = 0. It may be shown in the same manner that C± = 0, €5 = 0, • • • .
The coefficient Ci is zero also, since the product xXn does not contain x". Finally,
to find Co and C2 we need only equate the coefficients of xn + 1 and then equate
the two sides for x = 1. Doing this, we obtain the recurrent formula
(22) (n + l)Xn + l - (2n + l)xXn + nX,,_i = 0,
which affords a simple means of calculating the polynomials Xn successively.
The relation (22) shows that the sequence of polynomials
/oo\ ~TT ~V ~Y~ ~y
\£&) -^-Oi -**-!} -"-2i ' " i -"-n
possesses the properties of a Sturm sequence. As x varies continuously from — 1
to + 1, the number of changes of sign in this sequence is unaltered except when
x passes through a root of Xn = 0. But the formulse (19) show that there are n
changes of sign in the sequence (23) f or x = — 1, and none for x = 1. Hence
the equation Xn = 0 has n real roots between — 1 and -f 1, which also readily
follows from Rolle's theorem.
IV, §89] IMPROPER AND LINE INTEGRALS 175
IV. GENERALIZATIONS OF THE IDEA OF AN INTEGRAL
IMPROPER INTEGRALS LINE INTEGRALS*
89. The integrand becomes infinite. Up to the present we have sup
posed that the integrand remained finite between the limits of inte
gration. In certain cases, however, the definition may be extended
to functions which become infinite between the limits. Let us first
consider the following particular case : f(x) is continuous for every
value of x which lies between a and b, and for x = b, but it becomes
infinite for x = a. We will suppose for definiteness that a < b.
Then the integral of f(x) taken between the limits a + e and
b (e > 0) has a definite value, no matter how small e be taken. If
this integral approaches a limit as e approaches zero, it is usual and
natural to denote that limit by the symbol
Jf
/(*) dx.
If a primitive of /(cc), say F(x~), be known, we may write
C
Ja +
and it is sufficient to examine F(a -f- e) for convergence toward a
limit as c approaches zero. We have, for example,
Mdx
r
lL
If fj. > 1, the term l/c^"1 increases indefinitely as e approaches zero.
But if /u, is less than unity, we may write l/e?~l = e1"'1, and it is
clear that this term approaches zero with c. Hence in this case
the definite integral approaches a limit, and we may write
C" Mdx
I Tr ^~
Ja (X — a)
If fi = 1, we have;
/:
«/a4
M dx (b — a
— = M log
and the right-hand side increases indefinitely when e approaches zero.
To sum up, the necessary and sufficient condition that the given inte
gral should approach a limit is that /x should be less than unity.
*It is possible, if desired, to read the next chapter before reading the closing sec
tions of this chapter.
176 DEFINITE INTEGRALS [IV, §89
The straight line x = a is an asymptote of the curve whose equa
tion is
U
if p. is positive. It follows from the above that the area bounded by
the x axis, the fixed line x = b, the curve, and its asymptote, has a
finite value provided that //,<!.
If a primitive of f(x) is not known, we may compare the given
integral with known integrals. The above integral is usually taken
as a comparison integral, which leads to certain practical rules which
are sufficient in many cases. In the first place, the upper limit b
does not enter into the reasoning, since everything depends upon the
manner in which f(x} becomes infinite for x = a. We may therefore
replace b by any number whatever between a and b, which amounts
to writing f*+f = fa°+( + f* . In particular, unless f(x) has an infi
nite number of roots near x = a, we may suppose that f(x) keeps
the same sign between a and c.
We will first prove the following lemma :
Let $(x) be a function which is positive in the interval (a, b),
and suppose that the integral f* ^ <f> (x) dx approaches a limit as e
approaches zero. Then, if \f(x) \<<j>(x) throughout the whole inter
val, the definite integral fa + tf(x)dx also approaches a limit.
Hf(x) is positive throughout the interval (a, b), the demonstration
is immediate. For, since f(x) is less than </> (x), we have
/ f(x)dx < I ^(x}dx.
*J a + e J a + e
Moreover f*+ff(x)dx increases as c diminishes, since all of its ele
ments are positive. But the above inequality shows that it is con
stantly less than the second integral ; hence it also approaches a
limit. If f(x} were always negative between a and b, it would
be necessary merely to change the sign of each element. Finally,
if the function f(x) has an infinite number of roots near x = a, we
may write down the equation
f f(x) dx = f [/(*) + | /(or) \\dx-f \f(x) | dx.
*Ja + f c/a + e <Ja + s
The second integral on the right approaches a limit, since
|/(*)| <$(*). Now the function f(x) + \f(x)\ is either positive
IV, §89] IMPROPER AND LINE INTEGRALS 177
or zero between a and b, and its value cannot exceed 2 <£(#); hence
the integral
f
i/a +
also approaches a limit, and the lemma is proved.
It follows from the above that if a function f(x) does not approach
any limit whatever for x = a, but always remains less than a fixed
number, the integral approaches a limit. Thus the integral
fQl sin(l/x)dx has a perfectly definite value.
Practical rule. Suppose that the function /(#) can be written in
the form
where the function ij/(x) remains finite when x approaches a.
If fjL < 1 and the function \]/ (x~) remains less in absolute value than
a fixed number M, the integral approaches a limit. But if /JL ^ 1 and
the absolute value of ty(x) is greater than a positive number ra, the
integral approaches no limit.
The first part of the theorem is very easy to prove, for the abso
lute value of f(x) is less than M/(x — aY, and the integral of the
latter function approaches a limit, since p, < 1.
In order to prove the second part, let us first observe that ^(a*)
keeps the same sign near x = a, since its absolute value always
exceeds a positive number m. We shall suppose that \(/ (x) > 0
between a and b. Then we may write
X
4-
m dx
and the second integral increases indefinitely as e decreases.
These rules are sufficient for all cases in which we can find an
exponent p such that the product (x — ^/(.i") approaches, for
x = a, a limit A' different from zero. If /* is less than unity, the
limit b may be taken so near a that the inequality
holds inside the interval (a, J), where L is a positive number greater
178 DEFINITE INTEGRALS [IV, §89
than I K \. Hence the integral approaches a limit. On the other hand,
if p ^ 1, b may be taken so near to a that
I./ V"V I •" /,r n\t>.
\'c a)
inside the interval (a, £>), where I is a positive number less than | K\.
Moreover the function f(x), being continuous, keeps the same sign ;
hence the integral fb f(x)dx increases indefinitely in absolute
value.*
Examples. Let/(x) = P/Q be a rational function. If a is a root
of order m of the denominator, the product (x — a)m/(x) approaches
a limit different from zero for x = a. Since m is at least equal to
unity, it is clear that the integral f f(x)dx increases beyond all
limit as e approaches zero. But if we consider the function
/(*) =
where P and R are two polynomials and -R(ar) is prime to its deriv
ative, the product (x — a)1/2/(z) approaches a limit for x = a if a
is a root of R(x), and the integral itself approaches a limit. Thus
the integral
dx
f
J -\
approaches 7r/2 as e approaches zero.
Again, consider the integral ffl \ogxdx. The product #1/2loga;
has the limit zero. Starting with a sufficiently small value of x, we
may therefore write log x < Mx~ 1/2, where M is a positive number
chosen at random. Hence the integral approaches a limit.
Everything which has been stated for the lower limit a may be
repeated without modification for the upper limit b. If the function
f (cc) is infinite for x — b,we would define the integral Ja /(#) dx to be
the limit of the integral /J*"'/ (x) dx as c' approaches zero. If /(#)
is infinite at each limit, we would define f f (x~) dx as the limit of
the integral Cb"ef(x)dx as c and e' both approach zero independ
ently of each other. Let c be any number between a and b. Then
we may write
*The first part of the proposition may also be stated as follows: the integral has
a limit if an exponent /JL can be found (0 < /* < 1) such that the product (x — a)>*f(x)
approaches a limit A as a; approaches a, — the case where A = 0 not being excluded.
IV, §90] IMPROPER AND LINE INTEGRALS 179
r /(*)<& . f f(x)dx + c
U a + f <J a- + f *J c
and each of the integrals on the right should approach a limit in
this case.
Finally, if f(x) becomes infinite for a value c between a and b,
we would define the integral §*/(%) dx as the sum of the limits of
the two integrals f°~ef(x)dx, fcb+ff(x)dx, and we would proceed
in a similar manner if any number of discontinuities whatever lay
between a and b.
It should be noted that the fundamental formula (6), which was
established under the assumption that f(x) was continuous between
a and b, still holds when f(x) becomes infinite between these limits,
provided that the primitive function F(x~) remains continuous. For
the sake of definiteness let us suppose that the f unction f(x) becomes
infinite for just one value c between a and b. Then we have
I f(x) dx = lim I f(x) dx + lim I f(x) dx ;
t) a e'=0t/a e = 0c/c + e
and if F(x") is a primitive of /(x), this may be written as follows :
Xft
f(x) dx = lim F(c - c') - F(a) + F(b) - lim F(c + e).
€'=0 f=0
Since the function F(x) is supposed continuous for x = c, F(c + e)
and F(c — e') have the same limit F(c), and the formula again
becomes
f
I j (xj ix
*J a
The following example is illustrative :
J + 1 dx
-J-
i x
If the primitive function F(x) itself becomes infinite between a and
b, the formula ceases to hold, for the integral on the left has as yet
no meaning in that case.
The formulae for change of variable and for integration by parts
may be extended to the new kinds of integrals in a similar manner
by considering them as the limits of ordinary integrals.
90. Infinite limits of integration. Let/(x) be a function of x which
is continuous for all values of x greater than a certain number «.
Then the integral f lf(x) dx, where I > a, has a definite value, no
180 DEFINITE INTEGRALS [IV, §90
matter how large I be taken. If this integral approaches a limit
as I increases indefinitely, that limit is represented by the symbol
f
f(x)dx.
If a primitive of f(x) be known, it is easy to decide whether the
integral approaches a limit. For instance, in the example
dx
f
Jo
= arc tan I
the right-hand side approaches Tr/2 as I increases indefinitely, and
this is expressed by writing the equation
/
Ja
7T
2'
Likewise, if a is positive and //, — 1 is different from zero, we have
' kdx k / 1 1
I
c/a
If /A is greater than unity, the right-hand side approaches a limit as
I increases indefinitely, and we may write
° kdx k
On the other hand, if /i is less than one, the integral increases indefi
nitely with I. The same is true for p. = 1, for the integral then
results in a logarithm.
When no primitive of /(#) is known, we again proceed by com
parison, noting that the lower limit a may be taken as large as we
please. Our work will be based upon the following lemma :
Let <f> (x) be a function which is positive for x > a, and suppose that
the integral JJ <f> (x) dx approaches a limit. Then the integral f lf(x) dx
also approaches a limit provided that \f(x) \ ^ <£ («) for all values of
x greater than a.
The proof of this proposition is exactly similar to that given above.
If the function f(x) can be put into the form
/(») = *<
where the function ty(x) remains finite when x is infinite, the follow
ing theorems can be demonstrated, but we shall merely state them
IV, §91] IMPROPER AND LINE INTEGRALS 181
If the absolute value of ^ (x~) is less than a fixed number M and
p. is greater than unity, the integral approaches a limit.
If the absolute value of (]/ (x) is greater than a positive number m
and p. is less than or equal to unity, the integral approaches no limit.
For instance, the integral
/
cos ax
iT^dx
approaches a limit, for the integrand may be written
cos ax 1 cos ax
i+*°-**rT|'
and the coefficient of 1/x2 is less than unity in absolute value.
The above rule is sufficient whenever we can find a positive num
ber p, for which the product x*f(x) approaches a limit different from
zero as x becomes infinite. The integral approaches a limit if p, is
greater than unity, but it approaches no limit if p. is less than or
equal to unity.*
For example, the necessary and sufficient condition that the inte
gral of a rational fraction approach a limit when the upper limit
increases indefinitely is that the degree of the denominator should
exceed that of the numerator by at least two units. Finally, if we
take
where P and R are two polynomials of degree p and r, respectively,
the product xr/2~"f(x) approaches a limit different from zero when
x becomes infinite. The necessary and sufficient condition that the
integral approach a limit is that p be less than r/2 — 1.
91. The rules stated above are not always sufficient for determin
ing whether or not an integral approaches a limit. In the example
f(x) = (sin x)/x} for instance, the product x*f(x) approaches zero if
p. is less than one, and can take on values greater than any given
number if p, is greater than one. If p. = 1, it oscillates between + 1
and — 1. None of the above rules apply, but the integral does ap
proach a limit. Let us consider the slightly more general integral
* The integral also approaches a limit if the product x^f(x) (where M> 1) approaches
zero as x becomes infinite.
182 DEFINITE INTEGRALS [IV, §91
/i
-ax S1
e-~-
sin a; ,
dx, a>0.
The integrand changes sign for x = kir. We are therefore led to
study the alternating series
(24) «0 - «i + «2 - as + ... + (_ !)»«„ + . . .,
where the notation used is the following:
/» I
/
a0= I
Jo
» 27T •
sm x [ sin cc ,
e~ax -- -dx,
sin x
BIT X
Substituting y + nir for x, the general term an may be written
y + WT
It is evident that the integrand decreases as n increases, and hence
an + i<an- Moreover the general term an is less than f*(l/mr)dy,
that is, than 1/n. Hence the above series is convergent, since the
absolute values of the terms decrease as we proceed in the series,
and the general term approaches zero. If the upper limit I lies
between mr and (n + 1) TT, we shall have
-dx = Sn±6an, 0<9<1,
where Sn denotes the sum of the first n terms of the series (24). As
I increases indefinitely, n does the same, an approaches zero, and the
integral approaches the sum S of the series (24).
In a similar manner it may be shown that the integrals
r+* r+
I sino;2^x, I
Jo Jo
which occur in the theory of diffraction, each have finite values.
The curve y = sin a;2, for example, has the undulating form of a sine
curve, but the undulations become sharper and sharper as we go out,
since the difference ^/(n + I)TT — •^/n7^ of two consecutive roots of
sin x2 approaches zero as n increases indefinitely.
Remark. This last example gives rise to an interesting remark. As x increases
indefinitely sin «2 oscillates between — 1 and + 1. Hence an integral may
approach a limit even if the integrand does not approach zero, that is, even if
IV, §92J IMPROPER AND LINE INTEGRALS 183
the x axis is not an asymptote to the curve y = /(x). The following is an example
of the same kind in which the function /(x) does not change sign. The function
l + x6sin2x
remains positive when x is positive, and it does not approach zero, since
f(kn) = kit. In order to show that the integral approaches a limit, let us con
sider, as above, the series
flo + <*i + • • • + o« + • • • i
where
a« =
l + x6sin2x
As x varies from nit to (n + 1) TT, x6 is constantly greater than n6 7t6, and we may
write
l)rr dx
A primitive function of the new integrand is
== arc tan ( V 1 + n° n6 tan x),
f n*7t6
and as x varies from mt to (n + 1) TT, tan x becomes infinite just once, passing
from + co to — oo. Hence the new integral is equal (§ 77) to 7T/V1+ ns7f6, and
we have
*2 (n + 1)
— - -
n , — _
Vl + n« TT" n3 it
It follows that the series 2a« is convergent, and hence the integral J^ /(x) dx
approaches a limit.
On the other hand, it is evident that the integral cannot approach any limit
if /(x) approaches a limit h different from zero when x becomes infinite. For
beyond a certain value of x, /(x) will be greater than | h/2 \ in absolute value
and will not change sign.
The preceding developments bear a close analogy to the treatment of infinite
series. The intimate connection which exists between these two theories is
brought out by a theorem of Cauchy's which will be considered later (Chapter
VIII). We shall then also find new criteria which will enable us to determine
whether or not an integral approaches a limit in more general cases than those
treated above.
92. The function T(a). The definite integral
(25) T(a)= f + °V-1e-*dx
Jo
has a determinate value provided that a is positive.
For, let us consider the two integrals
ri ri
I xa-'e-xdx, I ««-l«-*<fe,
184 DEFINITE INTEGRALS
[IV, § 93
where t is a very small positive number and I is a very large positive number.
The second integral always approaches a limit, for past a sufficiently large value
of x we have xa-le~x< 1/x2, that is, ex>xa + l. As for the first integral, the
product x1 - af(x) approaches the limit 1 as x approaches zero, and the necessary
and sufficient condition that the integral approach a limit is that 1 - a be less
than unity, that is, that a be positive. Let us suppose this condition satisfied.
Then the sum of these two limits is the function T(a), which is also called Euler's
integral of the second kind. This function T(a) becomes infinite as a approaches
zero, it is positive when a is positive, and it becomes infinite with a. It has
a minimum for 0 = 1.4616321-.., and the corresponding value of T(a) is
0. 8856032- • -.
Let us suppose that a> 1, and integrate by parts, considering e~xdx as the
differential of — erx. This gives
but the product xa~le-x vanishes at both limits, since a > 1, and there remains
only the formula
(26) r(o) = (a - l)T(o - 1).
The repeated application of this formula reduces the calculation of Y(a) to
the case in which the argument a lies between 0 and 1. Moreover it is easy to
determine the value of T(a) when a is an integer. For, in the first place,
and the foregoing formula therefore gives, for a = 2, 3,
and, in general, if n is a positive integer,
(27) r(n) = 1.2.3...(n-l) = (n-l)l.
93. Line integrals. Let AB be an arc of a continuous plane curve,
and let P (x, y) be a continuous function of the two variables x and
y along AB, where x and y denote the coordinates of a point of AB
with respect to a set of axes in its plane. On the arc AB let us
take a certain number of points of division mli mz, • • •, m{, • • •, whose
coordinates are (xlt y^, (x2, yz), •••, (xi; y.), • • ., and then upon each
of the arcs mi_^rni let us choose another point n{ (£,., ^.) at random.
Finally, let us consider the sum
(28)
, *,*! -*,_, -..
extended over all these partial intervals. When the number of points
of division is increased indefinitely in such a way that each of the
differences xi — xi_l approaches zero, the above sum approaches a
IV, § 93]
IMPROPER AND LINE INTEGRALS
185
limit which is called the line integral of P(x, y) extended over the
arc AB, and which is represented by the symbol
JAB
P(x, y)dx.
In order to establish the existence of this limit, let us first sup
pose that a line parallel to the y axis cannot meet the arc AB in
more than one point. Let a and b be the abscissae of the points A
and B, respectively, and let y = <f>(x) be the equation of the curve AB.
Then <£(#) is a continuous function of x in the interval (a, b), by
hypothesis, and if we replace y by <f>(x~) in the function P(x, y), the
resulting function $(cc) = P[x, <£(X)] is also continuous. Hence we
have
and the preceding sum may therefore be written in the form
*(£,) to - a)
It follows that this surn approaches as its limit the ordinary definite
integral
I &(x)dx= I P[x, <t>(x)~\dx,
i/a t/a
and we have finally the formula
I P(x, y}dx = I P[x, t(x)-]dx.
JAB Ja
If a line parallel to the y axis can meet the arc AB in more than
one point, we should divide the arc
into several portions, each of which
is met in but one point by any line
parallel to the y axis. If the given
arc is of the form A CDB (Fig. 14),
for instance, where C and D are
points at which the abscissa has an
extremum, each of the arcs A C, CD,
DB satisfies the above condition, and
we may write
I P(x, y)dx= I P(x, y)dx + I P(x, y)dx + f P(x,y)dx.
JACDB J,T JCD JOB
But it should be noticed that in the calculation of the three integrals
FIG. 14
186 DEFINITE INTEGRALS [IV, §93
on the right-hand side the variable y in the function P(x, y)
must be replaced by three different functions of the variable x,
respectively.
Curvilinear integrals of the form JAR Q(x, y)dy may be denned
in a similar manner. It is clear that these integrals reduce at once
to ordinary definite integrals, but their usefulness justifies their
introduction. We may also remark that the arc AB may be com
posed of portions of different curves, such as straight lines, arcs of
circles, and so on.
A case which occurs frequently in practice is that in which the
coordinates of a point of the curve AB are given as functions of a
variable parameter
where <j>(t) and \(/(t), together with their derivatives <£'(£) and
are continuous functions of t. We shall suppose that as t varies
from a to ft the point (x, y) describes the arc AB without changing
the sense of its motion. Let the interval (a, /?) be divided into a
certain number of subintervals, and let ti_l and tf be two consecu
tive values of t to which correspond, upon the arc AB, two points
m,...! and m{ whose coordinates are (#,•_!, yf_i) and (x,-, yt), respec
tively. Then we have
where 0,. lies between ti_1 and t{. To this value 0, there corresponds
a point (£,-, 17,) of the arc mi_lmi; hence we may write
or, passing to the limit,
P(x,
f
»/.4
An analogous formula for JQdy may be obtained in a similar manner.
Adding the two, we find the formula
(29) f P<& -f Qdy = f
J^l£ »/«
which is the formula for change of variable in line integrals. Of
course, if the arc AB is composed of several portions of different
curves, the functions <f>(t) and \fr(£) will not have the same form
along the whole of AB, and the formula should be applied in that
case to each portion separately.
IV, §t)4] IMPROPER AND LINE INTEGRALS 187
94. Area of a closed curve. We have already defined the area of a
portion of the plane bounded by an arc A MB, a straight line which
does not cut that arc, and the two perpendiculars AAQ, BB0 let fall
from the points A and B upon the straight line (§§ 65, 78, Fig. 9).
Let us now consider a continuous closed curve of any shape, by
which we shall understand the locus described by a point M whose
coordinates are continuous functions x =f(£), y = <f>(t) of a param
eter t which assume the same values for two values t0 and T of
the parameter t. The functions f(f) and <j>(t~) may have several
distinct forms between the limits t0 and T; such will be the case,
for instance, if the closed contour C be composed of portions of
several distinct curves. Let M0, Mlt J/2, • • -, M{_ u Mif • • •, Mn_lt M0
denote points upon the curve C corresponding, respectively, to the
values t0, tl} t2, ••-, ti_l, tt, • • •, tn_ j, T of the parameter, which
increase from t0 to T. Connecting these points in order by straight
lines, we obtain a polygon inscribed in the curve. The limit
approached by the area of this polygon, as the number of sides is
indefinitely increased in such a way that each of them approaches
zero, is called the area of the closed curve C.* This definition is
seen to agree with that given in the particular case treated above.
For if the polygon A0A(21Q2 • • • BB0A0 (Fig. 9) be broken up into
small trapezoids by lines parallel to AA0, the area of one of these
trapezoids is (*,. - *,._,) [/(a-,-) + f(x{_ x)]/2, or (a\ -*<_,)/&),
where £,. lies between x^-^ and ccf. Hence the area of the whole
polygon, in this special case, approaches the definite integral
ff(x)dx.
Let us now consider a closed curve C which is cut in at most two
points by any line parallel to a certain fixed direction. Let us
choose as the axis of y a line parallel to this direction, and as the
axis of a; a line perpendicular to it, in such a way that the entire
curve C lies in the quadrant xOy (Fig. 15).
The points of the contour C project into a segment ab of the axis
Ox, and any line parallel to the axis of y meets the contour C in at
most two points, m^ and mz. Let yv = ^(cc) and ?/2 = tl/2(x) be the
equations of the two arcs AmvB and AmzB, respectively, and let
us suppose for simplicity that the points A and B of the curve C
which project into a and b are taken as two of the vertices of the
* It is supposed, of course, that the curve under consideration has no double point,
and that the sides of the polygon have been chosen so small that the polygon itself
has no double point.
188
DEFINITE INTEGRALS
[IV, § 94
polygon. The area of the inscribed polygon is equal to the differ
ence between the areas of the two polygons formed by the lines A a,
ab, bB with the broken lines inscribed in the two arcs Am2B and
AmiB, respectively. Passing to the limit, it is clear that the area
of the curve C is equal to the difference between the two areas
bounded by the contours Am^BbaA and Am^BbaA, respectively, that
is, to the difference between
the corresponding definite in
tegrals
X6 /tb
\l/z(x)dx — I ^
*J a
FIG. 15
These two integrals represent
the curvilinear integral fydx
taken first along Am2B and
then along Am-^B. If we
agree to say that the contour
C is described in the positive
sense when an observer standing upon the plane and walking around
the curve in that sense has the enclosed area constantly on his left
hand (the axes being taken as usual, as in the figure), then the above
result may be expressed as follows : the area O enclosed by the
contour C is given by the formula
(30)
r
J(C)
where the line integral is to be taken along the closed contour C in
the positive sense. Since this integral is unaltered when the origin
is moved in any way, the axes remaining parallel to their original
positions, this same formula holds whatever be
the position of the contour C with respect to
the coordinate axes.
Let us now consider a contour C of any form
whatever. We shall suppose that it is possible
to draw a finite number of lines connecting
pairs of points on C in such a way that the
resulting subcontours are each met in at most
two points by any line parallel to the y axis.
Such is the case for the region bounded by the
contour C in Fig. 16, which we may divide into three subregions
bounded by the contours amba, abndcqa, cdpc, by means of the
FIG. 16
IV, § 95]
IMPROPER AND LINE INTEGRALS
189
transversals ab and cd. Applying the preceding formula to each
of these subregions and adding the results thus obtained, the line
integrals which arise from the auxiliary lines ab and cd cancel each
other, and the area bounded by the closed curve C is still given by
the line integral —fydx taken along the contour C in the positive
sense.
Similarly, it may be shown that this same area is given by the
formula
(31)
n = I
J(C
x dy\
and finally, combining these two formulae, we have
(32)
«=- f
2 J<c
xdy — ydx,
where the integrals are always taken in the positive sense. This
last formula is evidently independent of the choice of axes.
If, for instance, an ellipse be given in the form
its area is
x = a cos t, y = b sin t,
1 C2"
fi = — I ab(cos2t -{- sm2f)dt = Trab.
2 Jo
95. Area of a curve in polar coordinates. Let us try to find the
area enclosed by the contour OAMBO (Fig. 17), which is composed
of the two straight lines OA, OB, and the arc A MB, which is
met in at most one
point by any radius ,,-—/ -^37'
vector. Let us take
0 as the pole and a
straight line Ox as /
the initial line, and I
let p = /(o>) be the \
equation of the arc
A MB.
Inscribing a polygon in the arc A MB, with A and B as two of
the vertices, the area to be evaluated is the limit of the sum of such
triangles as OMM'. But the area of the triangle OMM' is
FIG. 17
1 P
- p(p + Ap) sin Aw = Aw I
190 DEFINITE INTEGRALS [IV, §95
where « approaches zero with Aw. It is easy to show that all the
quantities analogous to c are less than any preassigried number rj
provided that the angles Aw are taken sufficiently small, and that
we may therefore neglect the term cAw in evaluating the limit.
Hence the area sought is the limit of the sum 2p2 Aw/2, that is, it
is equal to the definite integral
where wt and w2 are the angles which the straight lines OA and OB
make with the line Ox.
An area bounded by a contour of any form is the algebraic sum
of a certain number of areas bounded by curves like the above. If
we wish to find the area of a closed contour surrounding the point
0, which is cut in at most two points by any line through 0, for
example, we need only let w vary from 0 to 2?r. The area of a con
vex closed contour not surrounding O (Fig. 17) is equal to the dif
ference of the two sectors 0AM BO and OANBO, each of which may
be calculated by the preceding method. In any case the area is
represented by the line integral
taken over the curve C in the positive sense. This formula does
not differ essentially from the previous one. For if we pass from
rectangular to polar coordinates we have
x = p cos w, y — p &m w>
dx = cos w dp — p sin w c?w, dy = sin w dp + p cos w c?w,
x dy — y dx = p2dta.
Finally, let us consider an arc AMB whose equation in oblique
coordinates is y =f(x~). In order to find the area bounded by this
arc AMB, the x axis, and the two lines AA0, BB0, which are parallel
to the y axis, let us imagine a polygon inscribed in the arc AMB, and
let us break up the area of this polygon into small trapezoids by
lines parallel to the y axis. The area of one of these trapezoids is
IV, § 96]
IMPROPER AND LINE INTEGRALS
191
which may be written in the form (z,--i ~ #,•)/(£•) sin 0, where |
lies in the interval (xt_l; x^. Hence the area in question is equal
to the definite integral
sin $£ f(i
x) dx,
where x0 and A" denote the abscissae
of the points A and B, respectively.
It may be shown as in the similar
case above that the area bounded by
any closed contour C whatever is given
by the formula
A0 B0
FIG. 18
x dy — y dx.
(O
Note. Given a closed curve C (Fig. 15), let us draw at any point
M the portion of the normal which extends toward the exterior,
and let a, ft be the angles which this direction makes with the axes
of x and y, respectively, counted from 0 to TT. Along the arc Am^B
the angle ft is obtuse and dx = — ds cos ft. Hence we may write
I y dx = — \ y cos ft ds.
\J (Am^B) J
Along BmzA the angle ft is acute, but dx is negative along Bm2A
in the line integral. If we agree to consider ds always as positive,
we shall still have dx = — ds cos ft. Hence the area of the closed
curve may be represented by the integral
y cos ft ds,
where the angle ft is defined as above, and where ds is essentially
positive. This formula is applicable, as in the previous case, to a
contour of any form whatever, and it is also obvious that the same
area is given by the formula
x cos a ds.
These statements are absolutely independent of the choice of axes.
96. Value of the integral /xdy — ydx. It is natural to inquire what will
be represented by the integral fxdy — ydx, taken over any curve whatever,
closed or unclosed.
192 DEFINITE INTEGRALS [TV, §97
Let us consider, for example, the two closed curves OAOBO and
ApBqCrAsBtCuA (Fig. 19) which have one and three double points, respec
tively. It is clear that we may replace either of these curves by a combination
of two closed curves without double points. Thus the closed contour OA OBO
is equivalent to a combination of the
two contours 0-40 and OBO. The
integral taken over the whole contour
is equal to the area of the portion
0.40 less the area of the portion
OBO. Likewise, the other contour
may be replaced by the two closed
curves ApBqCrA and AsBtCuA, and
the integral taken over the whole con
tour is equal to the sum of the areas of ApBsA, BtCqB, and ArCuA, plus twice
the area of the portion AsBqCuA. This reasoning is, moreover, general. Any
closed contour with any number of double points determines a certain number
of partial areas <TI, <r2, • • •, <rp, of each of which it forms all the boundaries.
The integral taken over the whole contour is equal to a sum of the form
where mi, m2, • • • , mp are positive or negative integers which may be found by
the following rule : Given two adjacent areas <r, <r', separated by an arc ab of the
contour C, imagine an observer walking on the plane along the contour in the sense
determined by the arrows ; then the coefficient of the area at his left is one greater
than that of the area at his right. Giving the area outside the contour the coeffi
cient zero, the coefficients of all the other portions may be determined successively.
If the given arc AB is not closed, we may transform it into a closed curve by
joining its extremities to the origin, and the preceding formula is applicable to
this new region, for the integral fxdy — ydx taken over the radii vectores OA
and OB evidently vanishes.
V. FUNCTIONS DEFINED BY DEFINITE INTEGRALS
97. Differentiation under the integral sign. We frequently have to
deal with integrals in which the function tcr43e integrated depends
not only upon the variable of integration but also upon one or more
other variables which we consider as parameters. Let f(x, a) be a
continuous function of the two variables x and a when x varies from
x0 to X and a varies between certain limits «0and a^ We proceed
to study the function of the variable a which is defined by the
definite integral
>,r
= Cf(x,a)dx,
•Jxn
where a is supposed to have a definite value between a0 and alf and
where the limits x0 and X are independent of a.
IV, §97] FUNCTIONS DEFINED BY INTEGRALS 193
We have then
(33) F(a + Aa) - F(a) = f [/(*, a + Aa) -f(x, a)] dx.
JjT0
Since the function f(x, a) is continuous, this integrand may be made
less than any preassigned number c by taking Aa sufficiently small.
Hence the increment AF(a) will be less than e\X — x0 in absolute
value, which shows that the function F(a) is continuous.
If the function f(x, a) has a derivative with respect to a, let us
write
f(x, a + Aa) - f(x, a) = Aa [/. (x, «) + e] ,
where e approaches zero with Aa. Dividing both sides of (33) by
Aa, we find
and if -q be the upper limit of the absolute values of c, the absolute
value of the last integral will be less than ri\X — x0\. Passing to
the limit, we obtain the formula
(34) ^
da
In order to render the above reasoning perfectly rigorous we must
show that it is possible to choose Aa so small that the quantity c
will be less than any preassigned number rj for all values of x between
the given limits x0 and X. This condition will certainly be satisfied
if the derivative fa (x, a) itself is continuous. For we have from
the law of the mean
f(x, a + Aa) -f(x, a) = Aa/, (x, a + 0Aa), 0 < $ < 1,
and hence
If the function fa is continuous, this difference e will be less than 77
for any values of x and a, provided that | Aa j is less than a properly
chosen positive number h (see Chapter VI, § 120).
Let us now suppose that the limits X and xti are themselves func
tions of a. If A.Y and Aa:0 denote the increments which correspond
to an increment Aa, we shall have
194 DEFINITE INTEGRALS [IV, §97
F(a + Aa)- F(a) = f \_f(x, a + A,r) -f(x, a)
Jx0
f*X+&X
+ / /(a, a + Aa) dx
Jx
rxo + ^0
— I f(x, a +
Jx
or, applying the first law of the mean for integrals to each of the
last two integrals and dividing by Aa,
F(a 4- Aa) — F(«) _ C A'/(x, a 4- Aa-) —f(x, a) ,
Aa J, Aa
AT
-^/(^o + ^A^, nr + Aa).
As Aa- approaches zero the first of these integrals approaches the
limit found above, and passing to the limit we find the formula
(35) ^ =
which is the general formula for differentiation under the integral
sign.
Since a line integral may always be reduced to a sum of ordinary
definite integrals, it is evident that the preceding formula may be
extended to line integrals. Let us consider, for instance, the line
integral
= f P(x, y,
JAB
F(a) = I P(x, y, a} dx + Q(x, y, a) dy
JAB
taken over a curve AB which is independent of a. It is evident that
we shall have
F\a) = I Pa(x, y, a)dx + Qa(x, y,
JAB
where the integral is to be extended over the same curve. On the
other hand, the reasoning presupposes that the limits are finite and
that the function to be integrated does not become infinite between
the limits of integration. We shall take up later (Chapter VIII,
§ 175) the cases in which these conditions are not satisfied.
IV, §98] FUNCTIONS DEFINED BY INTEGRALS 195
The formula (35) is frequently used to evaluate certain definite
integrals by reducing them to others which are more easily calcu
lated. Thus, if a is positive, we have
/
Jo
1 *
— = arc tan —p.
va vo
whence, applying the formula (34) n — 1 times, we find
<-ir-.i:t...<.-i>r7S
Jo (^
98. Examples of discontinuity. If the conditions imposed are not satisfied for
all values between the limits of integration, it may happen that the definite inte
gral defines a discontinuous function of the parameter. Let us consider, for
example, the definite integral
f + 1 *«<***
J_l l-2xcosa + x2
This integral always has a finite value, for the roots of the denominator are
imaginary except when a = kit, in which case it is evident that F(a) = 0. Sup
posing that sin a ^ 0 and making the substitution x = cos a + t sin a, the indefi
nite integral becomes
/sin a dx f dt .
- - = I — = arc tan t.
1 - 2x cos a + x2 J 1 + t2
Hence the definite integral F(a) has the value
(1 — cos a\ I — 1 — cos a\
— ] — arc tan ( - ) »
sin a I \ sin a /
where the angles are to be taken between — n/2 and x/2. But
1 — cos a — 1 — cos a
- x — - = -1,
sin a sin a
and hence the difference of these angles is ± n/2. In order to determine the
sign uniquely we need only notice that the sign of the integral is the same as
that of sin a. Hence F(a) = ± n/2 according as sin a is positive or negative.
It follows that the function F(a) is discontinuous for all values of a of the form
kit. This result does not contradict the above reasoning in the least, however.
For when x varies from — 1 to + 1 and a varies from — e to + e, for example,
the function under the integral sign assumes an indeterminate form for the sets
of values a = 0, x = - 1 and a = 0, x = + 1 which belong to the region in ques
tion for any value of e.
It would be easy to give numerous examples of this nature. Again, consider
the integral
n -4- a:
sin mx
f
U—x
,
dx.
196 DEFINITE INTEGRALS [IV, §99
Making the substitution mx = y, we find
X
0
sin mx
y
where the sign to be taken is the sign of m, since the limits of the transformed
integral are the same as those of the given integral if m is positive, but should
be interchanged if m is negative. We have seen that the integral in the second
member is a positive number N (§ 91). Hence the given integral is equal to ± N
according as m is positive or negative. If m = 0, the value of the integral is
zero. It is evident that the integral is discontinuous for m = 0.
VI. APPROXIMATE EVALUATION OF DEFINITE INTEGRALS
99. Introduction. When no primitive of f(x) is known we may
resort to certain methods for finding an approximate value of the
definite integral f*f(x) dx. The theorem of the mean for integrals
furnishes two limits between which the value of the integral must
lie, and by a similar process we may obtain an infinite number of
others. Let us suppose that $(x) <f(x) < ty(x) for all values of x
between a and b (a < b). Then we shall also have
s*t> s*b /-i
I <j>(x}dx< I f(x)dx< I
<J a Ja c/a
If the functions <£(#)• and ^(a;) are the derivatives of two known
functions, this formula gives two limits between which the value of
the integral must lie. Let us consider, for example, the integral
C dx
"Jo Vl-a;4
Now Vl-z4 = Vl - x2 Vl + x\ and the factor Vl + z2 lies
between 1 and V2 for all values of x between zero and unity.
Hence the given integral lies between the two integrals
r} dx i rl dx
Jo Vl-cc2' V2Jo Vl— xa
that is, between Tr/2 and 7r/(2V2). Two even closer limits may
be found by noticing that (1 + z2)-1/2 is greater than 1 — x2/2,
which results from the expansion of (1 -f 7/)-1/2 by means of Taylor's
series with a remainder carried to two terms. Hence the integral
I is greater than the expression
dx 1 r} x2dx
IV, §y»] APPROXIMATE EVALUATION 197
The second of these integrals has the value Tr/4 (§ 105) ; hence /
lies between Tr/2 and 3 7r/8.
It is evident that the preceding methods merely lead to a rough
idea of the exact value of the integral. In order to obtain closer
approximations we may break up the interval (a, 6) into smaller
subintervals, to each of which the theorem of the mean for inte
grals may be applied. For definiteness let us suppose that the
function /(a:) constantly increases as x increases from a to b. Let
us divide the interval (a, b~) into n equal parts (b — a — nh). Then,
by the very definition of an integral, ^f(x}dx lies between the
two sums
s = h\f(a)
S = h\f(a + h) +f(a + 2A) -f •• • +/(a + nh)\.
If we take (S + s)/2 as an approximate value of the integral, the
error cannot exceed S-s\/2= [(i - «) /2 n] [/(i) -/(a)] . The
value of (S + s)/2 may be written in the form
(/(a) + /O
| ,
2
/•[a + (ro - 1) A] +f(a + nh) )
2 )
Observing that \f(a + ih) +/[a + (i +1) h~\\h/2 is the area of
the trapezoid whose height is h and whose bases are /(a + ih) and
f(a + ih + 7t), we may say that the whole method amounts to
replacing the area under the curve y = f(x) between two neighbor
ing ordinates by the area of the trapezoid whose bases are the two
ordinates. This method is quite practical when a high degree of
approximation is not necessary.
Let us consider, for example, the integral
/'
C/O
dx
Taking n = 4, we find as the approximate value of the integral
and the error is less than 1/16 = .0625.* This gives an approxi
mate value of TT which is correct to one decimal place, — 3.1311
* Found from the formula \S — s\/2. In fact, the error is about .00260, the exact
value being rr/4. — TRANS.
198 DEFINITE INTEGRALS [IV, § 100
If the function f(x) does not increase (or decrease) constantly as
x increases from a to b, we may break up the interval into sub-
intervals for each of which that condition is satisfied.
100. Interpolation. Another method of obtaining an approximate
value of the integral f f(x)dx is the following. Let us determine
a parabolic curve of order n,
y = <£(z) = a0 + a-iX + 1- anxn,
which passes through (n -f- 1) points B0, Bl} •••, Bn of the curve
y =f(x) between the two points whose abscissae are a and b.
These points having been chosen in any manner, an approximate
value of the given integral is furnished by the integral fb<j>(x}dx,
which is easily calculated.
Let (x0, T/O), (#!, 7/i), •••, (xn, ?/„) be the coordinates of the (n +1)
points BQ, B!, •••, Bn. The polynomial <f)(x) is determined by
Lagrange's interpolation formula in the form
<£(*) = Z/o A'0 -f ijt A*! -\ h i/i Xf -\ \-yn Xn ,
where the coefficient of yt is a polynomial of degree n,
x _ (^ - EQ) •••(x-xi_l}(x- xi + }} • • • (x - ay) ^
' (X- - Xo) • • • (*i ~ Xi- l) (X -*<+!)••• (*< - Xn) '
which vanishes for the given values x0, x±, • • • , xn, except for x = xf,
and which is equal to unity when x — x{. Hence we have
/'
U a
• Q « ' ' :
The numbers xi are of the form
x0 = a -f 00(b — a), a?! = a -f- $i(& — «)• • • •, #„ = a + #„(& — «)»
where 0 < 00 < 6t < • • • < 6H 5 1. Setting x = a + (b — a) t, the ap
proximate value of the given integral takes the form
(36) (b — a) (A"0y0 + K\y\ H h A'.y,,),
where Ki is given by the formula
KI = I 77 ^ ''"^^ — dt .
If we divide the main interval (a, b) into subintervals whose
ratios are the same constants for any given function /(x) whatever,
the numbers 00, 0U • • •, dn, and hence also the numbers K{, are inde
pendent of f(x). Having calculated these coefficients once for all,
IV, §101] APPROXIMATE EVALUATION 199
it only remains to replace y0, yi, •••, yn by their respective values
in the formula (36).
If the curve f(x) whose area is to be evaluated is given graph
ically, it is convenient to divide the interval (a, 6) into equal parts,
and it is only necessary to measure certain equidistant ordinates of
this curve. Thus, dividing it into halves, we should take 00 = 0,
61 = 1/2, 02 = 1, which gives the following formula for the approxi
mate value of the integral :
b — a
I = —Q- u/o + 4yi + y2) •
Likewise, for n = 3 we find the formula
1 = —£— C?/0 + 3#! + 3//2 + 7/s) ,
o
and for n = 4
7 = ~9o ~ (7z/o
Ir
The preceding method is due to Cotes. The following method,
due to Simpson, is slightly different. Let the interval (a, b~) be
divided into 2n equal parts, and let y0, y1} ?/2, •••, y2n be the ordi
nates of the corresponding points of division. Applying Cotes'
formula to the area which lies between two ordinates whose indices
are consecutive even numbers, such as y0 and y2, ?/2 and yt, etc., we
find an approximate value of the given area, in the form
whence, upon simplification, we find Simpson's formula :
1 = -- [2/0 + 2/2« + 2(y2 + y, H ---- -f ?/2n_2)
101. Gauss' method. In Gauss' method other values are assigned
the quantities #,. The argument is as follows: Suppose that we
can find polynomials of increasing degree which differ less and less
from the given integrand f(x~) in the interval (a, &). Suppose,
for instance, that we can write
/(*) = a0 + a,x + a2x* + • • • + a^ X2n~l + 7?2n(x) ,
where the remainder R2H(x~) is less than a fixed number ^ for all
200 DEFINITE INTEGRALS [IV, § 101
values of x between a and b* The coefficients a, will be in gen
eral unknown, but they do not occur in the calculation, as we shall
see. Let x0, x1} ••-, xn_1 be values of x between a and b, and let
<£(ce) be a polynomial of degree n — 1 which assumes the same
values as does /(ce) for these values of x. Then Lagrange's inter
polation formula shows that this polynomial may be written in the
form
where <f»m and tyk are at most polynomials of degree n — I. It is
clear that the polynomial <f>m(x) depends only upon the choice of
xo> xi>'"> xn-i- On the other hand, this polynomial <£,„(#) must
assume the same values as does xm for x = x0, x = xl} • • •, x = xa_l.
For, supposing that all the a's except am and also R^(x) vanish,
f(x) reduces to amxm and </>(a:) reduces to am$in(x). Hence the
difference xm — <f>m(x) must be divisible by the product
Pn(x) = (x - XQ) (x - a-i) • • • (x - a^).
It follows that a1'" — <f>m(x)= PnQm_n(x~), where Qm_n(x) is a poly
nomial of degree m — n, if ra > n ; and that xm — <J>m (x) = 0 if m < n — 1.
The error made in replacing f f(x~) dx by Ja <f> (x~) dx is evidently
given by the formula
^ r
•*"Mj"
_ V 7? (Y\ 1 \b (x
£4 H2,i (xi) 1 *i \x
i=0 ^o
The terms which depend upon the coefficients a0, a1, • • -, an_l vanish
identically, and hence the error depends only, upon the coefficients
<*«> an + i> '"> a-2n-\ an(l the remainder R2n(x'). But this remain
der is very small, in general, with respect to the coefficients
an, o-n + 1, •••, a.2n_l. Hence the chances are good for obtaining a
high degree of approximation if we can dispose of the quantities
x0, xlt •••, xn_l in such a way that the terms which depend upon
a«> an + i> •••> a2n-i a^so vanish identically. For this purpose it is
necessary and sufficient that the n integrals
s*b f*\> nb
/ PnQudx, I P^.dx, ••-, I
Ja Jo. fc/a
* This is a property of any function which is continuous in the interval (a,
according to a theorem due to Weierstrass (see Chapter IX, § 199).
IV, §102] APPROXIMATE EVALUATION 201
should vanish, where Q{ is a polynomial of degree i. We have
already seen (§ 88) that this condition is satisfied if we take Pn of
the form
**.-;£ [<*-«)• <*-*)•]-
It is therefore sufficient to take for x0, xi} •••, xn_l the n roots of
the equation Pn = 0, and these roots all lie between a and b.
We may assume that a = — 1 and b — + 1, since all other cases
may be reduced to this by the substitution x = (b + «) /2 + 2 (b — a) /2.
In the special case the values of x0, xlf ••-, «„_! are the roots of
Legendre's polynomial Xn. The values of these roots and the
values of Ki for the formula (36), up to n = 5, are to be found to
seven and eight places of decimals in Bertrand's Traite de Calcul
integral (p. 342).
Thus the error in Gauss' method is
C R*(x)dx -"j^R^x,) f *,
•Jo, t = 0 *^a
where the functions ^{(x) are independent of the given integrand.
In order to obtain a limit of error it is sufficient to find a limit of
R^(x), that is, to know the degree of approximation with which
the function f(x) can be represented as a polynomial of degree
2n — 1 in the interval (a, &). But it is not necessary to know
this polynomial itself.
Another process for obtaining an approximate numerical value of
a given definite integral is to develop the function f(x) in series and
integrate the series term by term. We shall see later (Chapter VIII)
under what conditions this process is justifiable and the degree of
approximation which it gives.
102. Amsler's planimeter. A great many machines have been invented to
measure mechanically the area bounded by a closed plane curve.* One of the
most ingenious of these is Amsler's planimeter, whose theory affords an interest
ing application of line integrals.
Let us consider the areas AI and A* bounded by the curves described by two
points AI and A2 of a rigid straight line which moves in a plane in any manner
arid finally returns to its original position. Let (xi, 2/1) and (x2, y2) be the coor
dinates of the points AI and A», respectively, with respect to a set of rectangu
lar axes. Let I be the distance AiA2, and 0 the angle which A\AZ makes with
* A description of these instruments is to be found in a work by Abdank-
Abakanowicx: Les integraphes, la courbe integrate et ses applications (Gauthier-
Villars, 1886).
2U2 DEFINITE INTEGRALS [IV, § 102
the positive x axis. In order to define the motion of the line analytically, «i, j/i,
and 6 must be supposed to be periodic functions of a certain variable parameter t
which resume the same values when t is increased by T. We have x2 = %i + I cos 6,
2/2 = V\ + I sin 0» an(l hence
— yidx\ + PdO
+ I(cos0dyi — sintfdxi +
The areas AI and A2 of the curves described by the points A\ and Az, under the
general conventions made above (§ 96), have the following values :
l r 1 r
AI = - J xidyi - yidxi, A2 = - J x2dy2 - yzdx2.
Hence, integrating each side of the equation just found, we obtain the equation
- CdO + - j fcos6dyi - s'm0dxl + f (xicosfl + yis\n0)d0\ ,
A2 = AI -f
where the limits of each of the integrals correspond to the values t0 and tQ + T
of the variable t. It is evident that fd8 = 2Kn, where K is an integer which
depends upon the way in which the straight line moves. On the other hand,
integration by parts leads to the formulae
/ Xi cos 6 d0 — Xi sin 0—1 sin 0 dx\ ,
f yi sin 0 d0 = — y\ cos 0 + / cos 9 dyi .
But Xi sin 0 and y\ cos 0 have the same values for t — tQ and t = to + T. Heuce
the preceding equation may be written in the form
A2 = AI + Knl2 + I CcosOdyi — si
Now let s be the length of the arc described by A i counted positive In a certain
sense from any fixed point as origin, and let a be the angle which the positive
direction of the tangent makes with the positive x axis. Then we shall have
cos 0 dyi — sin 6 dxi = (sin a cos 0 — sin 0 cos a) ds = sin V da ,
where V is the angle which the positive direction of the tangent makes with the
positive direction A^AZ of the straight line taken as in Trigonometry. The
preceding equation, therefore, takes the form
(38) A2 = AI + Kxt2 + ifsinVds.
Similarly, the area of the curve described by any third point A3 of the straight
line is given by the formula
(39) A3 = A! + K«l'* + I'fsiuVds,
where I' is the distance AiA&. Eliminating the unknown quantity fsinVds
between these two equations, we find the formula
1'Aj - IA» = (*' - 1) AI + Kxll\l - Hi
IV, §102] APPROXIMATE EVALUATION 203
which may be written in the form
(40) A! (23) + Ao (31 ) + A3 (12) + K* (12) (23) (31) = 0 ,
where (ik) denotes the distance between the points Ai and Ak (i, k = 1, 2, 8)
taken with its proper sign. As an application of this formula, let us consider
a straight line A\A<L of length (a + 6), whose extremities A\ and A2 describe the
same closed convex curve C. The point A3, which divides the line into seg
ments of length a and 6, describes a closed curve C" which lies wholly inside C.
In this case we have
A2 = Ai, (12) = a + b, (23) = - 6, (31) = - a, K = I •
whence, dividing by a + 6,
AI — A3 = rtab.
But AI — AS is the area between the two curves C and C'. Hence this area is
independent of the form of the curve C. This theorem is due to Holditch.
If, Instead of eliminating JsinFds between the equations (38) and (39), we
eliminate AI, we find the formula
(41) A3 = A2 + Kx(V* - I2) + (I' - l)CsinVds.
Amsler's planimeter affords an application of this formula. Let AiA2As be a
rigid rod joined at A2 with another rod (L42. The point 0 being fixed, the point
A3, to which is attached a sharp pointer, is made to describe the curve whose area
is sought. The point A% then
describes an arc of a circle or
an entire circumference, accord
ing to the nature of the motion.
In any case the quantities A2, -K",
I, I' are all known, and the area
AS can be calculated if the in
tegral Jsin Vds, which is to be
taken over the curve C\ described
by the point A\, can be evaluated.
f\ **
This end A\ carries a graduated V
, FIG. 20
circular cylinder whose axis coin
cides with the axis of the rod AiA3, and which can turn about this axis.
Let us consider a small displacement of the rod which carries AiA2A3 into
the position AiA'^Az. Let Q be the intersection of these straight lines. About
Q as center draw the circular arc Ai a and drop the perpendicular A{P from
AI upon AiA2. We may imagine the motion of the rod to consist of a sliding
along its own direction until AI comes to or, followed by a rotation about Q which
brings a to A{. In the first part of this process the cylinder would slide, with
out turning, along one of its generators. In the second part the rotation of
the cylinder is measured by the arc aA{. The two ratios aA\/A{P and
A{P/axcAiA{ approach 1 and sinF, respectively, as the arc A[A\ approaches
zero. Hence ccA{ = As (sinF + e), where e approaches zero with As. It follows
that the total rotation of the cylinder is proportional to the limit of the sum
SAs(sinF + e), that is, to the integral JsinFds. Hence the measurement of
this rotation is sufficient for the determination of the given area.
204 DEFINITE INTEGRALS [IV, Exs.
EXERCISES
1. Show that the sum 1/n + l/(n + !) + •••+ l/2n approaches log 2 as n
increases indefinitely.
[Show that this sum approaches the definite integral fQl [1/(1 + x)]dx as its
limit.]
2. As in the preceding exercise, find the limits of each of the sums
n + -5- + ..- +
n2 + 1 n2 + 22 n2 + (n - I)2
1 1 1
Vn2 - 1 Vn2 - 22 Vn* - (n - I)2
by connecting them with certain definite integrals. In general, the limit of
the sum
as n becomes infinite, is equal to a certain definite integral whenever <f>(i, n) is
a homogeneous function of degree — 1 in i and n.
3. Show that the value of the definite integral f0"/2 log sin x dx is
-(jr/2)log2.
[This may be proved by starting with the known trigonometric formula
. it . 2* . (n — \)it n
sm — sin sm - ^— — ,
n n n 2"-'
or else by use of the following almost self-evident equalities :
JT
/•I, fy. , 1 C'2. /sin2x\ _.
I log sin x dx = I log cos x dx = - log ( I dx. ]
«/o •'O 2 /u \ 2 /
4. By the aid of the preceding example evaluate the definite integral
1 tan x dx .
2/
5. Show that the value of the definite integral
,1
/
Jo
1 + x2
is (jr/8)log2.
[Set x = tan <p and break up the transformed integral into three parts.]
6*. Evaluate the definite integral
/ITT
I log (1 — 2a cos x + a2) dx .
Jo [POISSOK.]
IV, Exs.] EXERCISES 205
[Dividing the interval from 0 to it into n equal parts and applying a well-known
formula of trigonometry, we are led to seek the limit of the expression
?r ra — 1
-log - — «2n_!
n La + 1
as n becomes infinite. If a lies between — 1 and + 1, this limit is zero. If
a2 > 1, it is 7t log a2. Compare § 140.]
1. Show that the value of the definite integral
sinxdx
/'
Jo
/o Vl — 2or cos x + a2
where a is positive, is 2 if a < 1, and is 2/a if a > 1.
15*. Show that a necessary and sufficient condition that /(x) should be inte-
grable in an interval (a, b) is that, corresponding to any preassigned number e,
a subdivision of the interval can be found such that the difference S — s of the
corresponding sums S and s is less than e.
9. Let/(x) and <j>(x) be two functions which are continuous in the interval (a, 6),
and let (a, Xi, Xj, • • • , b) be a method of subdivision of that interval. If £,, 77.
are any two values of x in the interval (x,-_i, x,-), the sum 2/(£,-) <f> (?;,•) (x, — x,-_i)
approaches the definite integral f^f(x) <t>(x) dx as its limit.
10. Let/(x) be a function which is continuous and positive in the interval (a, b).
Show that the product of the two definite integrals
/>*' f/l>
is a minimum when the function is a constant.
11. Let the symbol I*1 denote the index of a function (§ 77) between
and Xi. Show that the following formula holds:
where e = + 1 if /(x0) > 0 and /(Xi) < 0, e = - 1 if f(x0) < 0 and f(xi) > 0, and
c = 0 if /(x0) and /(Xi) have the same sign.
[Apply the last formula in the second paragraph of § 77 to each of the func
tions /(x) and l//(x).]
12*. Let U and V be two polynomials of degree n and n — 1, respectively,
which are prime to each other. Show that the index of the rational fraction
V/U between the limits — oo and -f oo is equal to the difference between the
number of imaginary roots of the equation U + iV = 0 in which the coefficient
of i is positive and the number in which the coefficient of i is negative.
[HERMITE, Bulletin de la Socidte matMmatique, Vol. VII, p. 128.]
13*. Derive the second theorem of the mean for integrals by integration by
parts.
206 DEFINITE INTEGRALS [IV, Exs.
[Let/(x) and <f>(x) be two functions each of which is continuous in the inter
val (a, b) and the first of which, /(x), constantly increases (or decreases) and
has a continuous derivative. Introducing the auxiliary function
*(z) = f*<t>(x)dx
J a
and integrating by parts, we find the equation
f /(x) </>(x) dx = f(b) *(&) - f /'(x) *(x) dx .
va v a
Since /'(x) always has the same sign, it only remains to apply the first theorem
of the mean for integrals to the new integral.]
14. Show directly that the definite integral fxdy — ydx extended over a
closed contour goes over into an integral of the same form when the axes are
replaced by any other set of rectangular axes which have the same aspect.
15. Given the formula
/** 1
I cos Xx dx = - (sin \b — sin Xa) ,
J a X
evaluate the integrals
/^fc /ift
i x2P + 1sinXxdx, I x2^cosXxdx.
<J a J n
16. Let us associate the points (x, y) and (x', y') upon any two given curves
C and C", respectively, at which the tangents are parallel. The point whose
coordinates are x\ = px + qx', yi = py + qy', where p and q are given constants,
describes a new curve C\. Show that the following relation holds between the
corresponding arcs of the three curves :
Si = ± ps ± qs'.
17. Show that corresponding arcs of the two curves
c ( x = tf(t) - f(t) +t'(t), C'\x' = V'(t) - f(t) - *'(«) ,
have the same length whatever be the functions /(f) and <f>(t).
18. From a point M of a plane let us draw the normals -MPi, • • •, MPn to
n given curves Ci, C*2, • ••, Cn which lie in the same plane, and let k be the
distance MP(. The locus of the points M, for which a relation of the form
F(li, k) • • •) In) = 0 holds between the n distances Z,-, is a curve T. If lengths
proportional to cF/dli be laid off upon the lines MP^ respectively, according to
a definite convention as to sign, show that the resultant of these n vectors gives
the direction of the normal to F at the point M. Generalize the theorem for
surfaces in space.
19. Let C be any closed curve, and let us select two points p and p' upon the
tangent to C at a point TO, on either side of TO, making mp = mp'. Supposing
that the distance mp varies according to any arbitrary law as TO describes the
curve (7, show that the points p and p' describe curves of equal area. Discuss
the special case where mp is constant.
IV, Exs.] EXERCISES 207
20. Given any closed convex curve, let us draw a parallel curve by laying oft
a constant length I upon the normals to the given curve. Show that the area
between the two curves is equal to ± it I2 + s£, where s is the length of the given
curve.
21. Let C be any closed curve. Show that the locus of the points A, for
which the corresponding pedal has a constant area, is a circle whose center is
fixed.
[Take the equation of the curve C in the tangential form
x cos t + y sin t = /(£).]
22. Let C be any closed curve, C\ its pedal with respect to a point A, and C2
the locus of the foot of a perpendicular let fall from A upon a normal to C.
Show that the areas of these three curves satisfy the relation A = AI — A2.
[By a property of the pedal (§ 36), if p and u are the polar coordinates of a point
on d, the coordinates of the corresponding point of C2 are p' and u + n/2, and
those of the corresponding point of C are r = Vp2 + p"2 and <p = w + arc tan p'/p.]
23. If a curve C rolls without slipping on a straight line, every point A which
is rigidly connected to the curve C describes a curve which is called a roulette.
Show that the area between an arc of the roulette 'and its base is twice the area
of the corresponding portion of the pedal of the point A with respect to C. Also
show that the length of an arc of the roulette is equal to the length of the corre
sponding arc of the pedal. ro
[SXEINEK.]
[In order to prove these theorems analytically, let X and Y be the coordi
nates of the point A with respect to a moving system of axes formed of the
tangent and normal at a point M on C. Let s be the length of the arc OM
counted from a fixed point 0 on C, and let w be the angle between the tangents
at 0 and M. First establish the formulae
and then deduce the theorems from them.]
24*. The error made in Gauss' method of quadrature may be expressed in
the form
/(2n>(£) x 2 r 1.2.. 3... 71 -ft
1 . 2 • • • 2n 2n + 1 Ll . 2 • • • (2n - 1)J'
where £ lies between — 1 and +1. r,, .-, 100., ,
[MANSION, Comptes renews, 1886.]
CHAPTER V
INDEFINITE INTEGRALS
We shall review in this chapter the general classes of elemen
tary functions whose integrals can be expressed in terms of ele
mentary functions. Under the term elementary functions we shall
include the rational and irrational algebraic functions, the exponen
tial function and the logarithm, the trigonometric functions and
their inverses, and all those functions which can be formed by a
finite number of combinations of those already named. When the
indefinite integral of a function f(x) cannot be expressed in terms
of these functions, it constitutes a new transcendental function.
The study of these transcendental functions and their classification
is one of the most important problems of the Integral Calculus.
I. INTEGRATION OF RATIONAL FUNCTIONS
103. General method. Every rational function /(a:) is the sum of
an integral function E(x) and a rational fraction P(x~)/Q(x"), where
P(x) is prime to and of less degree than Q(#). If the real and
imaginary roots of the equation Q(x~) be known, the rational frac
tion may be decomposed into a sum of simple fractions of one or the
other of the two types
A MX + N
(cc-a)"1' [(x - a)2 + /32]»'
The fractions of the first type correspond to the real roots, those
of the second type to pairs of imaginary roots. The integral of
the integral function E(x) can be written down at once. The inte
grals of the fractions of the first type are given by the formulae
Adx A
(x-a)» (m-l)(a;-a)— >'
A dx
— = A log (x — a), if m = 1 .
x — a
For the sake of simplicity we have omitted the arbitrary constant C,
which belongs on the right-hand side. It merely remains to examine
208
V, §103] RATIONAL FLECTIONS 209
the simple fractions which arise from pairs of imaginary roots.
In order to simplify the corresponding integrals, let us make the
substitution
x = a -\- fit, dx — ftdt.
The integral in question then becomes
r MX + N 1 CMa + N+MQt
J [(x - a)* + W d' - 0*-' J (1+ *2)»
L-
and there remain two kinds of integrals :
c tat r dt
J (i + t'2f J (i + «a)"'
Since tdt is half the differential of 1 + t2, the first of these inte
grals is given, if n > 1, by the formula
r
J
tdt i _ __ /?2
or, if n — 1, by the formula
tdt 1
The only integrals which remain are those of the type
dt
r
J
If n = 1, the value of this integral is
dt x — a
= arc tan t = arc tan — - —
1 + t*
If n is greater than unity, the calculation of the integral may be
reduced to the calculation of an integral of the same form, in which
the exponent of (1 -f- £2) is decreased by unity. Denoting the inte
gral in question by /„, we may write
/i -j- 12 — tz r
~7T 5T~ dt = I '
(1 + t )" J <
From the last of these integrals, taking
tdt
210 INDEFINITE INTEGRALS [v, § 103
and integrating by parts, we find the formula
C t*dt __ t __ 1 C
J (i+O"~ 2(n-l)(l+<V-1 2(»-l)J
dt
Substituting this value in the equation for 7n, that equation becomes
_ 2n - 3 t
n o« O *n-l I
2n - 2 ' ' 2 (n -!)(! + *2)"~
Repeated applications of this formula finally lead to the integral
= arc tan t. Retracing our steps, we find the formula
(2n - 3) (2n - 5) - • • 3 . 1
' '
where 72 (£) is a rational function of £ which is easily calculated.
We will merely observe that the denominator is (1 + £2)"~1, and that
the numerator is of degree less than 2n — 2 (see § 97, p. 192).
It follows that the integral of a rational function consists of
terms which are themselves rational, and transcendental terms of
one of the following forms :
y _ f-f
log (SB — a), log [(x - a)2 + /8s], arc tan — - —
P
Let us consider, for example, the integral /[l/(#4 — 1)] dx. The
denominator has two real roots -f 1 and — 1, and two imaginary
roots + i and — i. We may therefore write
1 A B Cx + 7)
'
x* - 1 ~ x-l Z + l
In order to determine A, multiply both sides by x — 1 and then set
x = 1. This gives A = 1/4, and similarly B = — 1/4. The iden
tity assumed may therefore be written in the form
r.r + D
or, simplifying the left-hand side,
-1 _ Cx + D
2(1 + x3) ~ 1 + x2 '
It follows that C = 0 and 7) = — 1/2, and we have, finally,
1 1 1 1
•T" A / ~t "I \ .^ /'V — 1-.- 11 */ ( nr* \ 1 ^
«*^ -L V t •!/ -Ly Trl*^ ^^ / \ 1^ /
which gives
rfa; 1 , (x - 1\ 1
V, §104] RATIONAL FUNCTIONS 211
Note. The preceding method, though absolutely general, is not
always the simplest. The work may often be shortened by using
a suitable device. Let us consider, for example, the integral
dx
2 - 1)"
If n > 1, we may either break up the integrand into partial frac
tions byliieans of the roots + 1 and — 1, or we may use a reduction
formula similar to that for /„. But the most elegant method is to
make the substitution x = (1 -f «)/(! — 2), which gives
4* 2dz
dx =
(!-*)« (I-*)'
/7 O /"* /1 ~\2» — 2
(*» - 1)- = 4"- J " *» rf*'
Developing (1 — z)2n~2 by the binomial theorem, it only remains
to integrate terms of the form Az*t where \L may be positive or
negative.
104. Hermite's method. We have heretofore supposed that the
fraction to be integrated was broken up into partial fractions, which
presumes a knowledge of the roots of the denominator. The fol
lowing method, due to Hermite, enables us to find the algebraic
part of the integral without knowing these roots, and it involves
only elementary operations, that is to say, additions, multiplications,
and divisions of polynomials.
Let f(x)/F(x) be the rational fraction which is to be integrated.
We may assume that f(x) and F(x) are prime to each other, and
we may suppose, according to the theory of equal roots, that the
polynomial F(x) is written in the form
where Xl} A'2, •••, A'^ are polynomials none of which have multiple
roots and no two of which have any common factor. We may now
break up the given fraction into partial fractions whose denomina
tors are Xlt X\, •••, Xpp:
X\ X*
where ^4, is a polynomial prime to Xt. For, by the theory of high
est common divisor, if X and Y are any two polynomials which are
212 INDEFINITE INTEGRALS [v, § 104
prime to each other, and Z any third polynomial, two other poly
nomials A and B may always be found such that
BX + AY= Z.
Let us set X = Xlt Y = X\ • • • Xpp, and Z =/(*). Then this identity
becomes
BXi + AX\--.Xl=f(x),
or, dividing by F(x),
It also follows from the preceding identity that if f(x) is prime to
F(x~), A is prime to X1 and B is prime to X\ • • • X*. Kepeating the
process upon the fraction
B
and so on, we finally reach the form given above.
It is therefore sufficient to show how to obtain the rational part
of an integral of the form
/A dx
~¥"
where <£(.x) is a polynomial which is prime to its derivative. Then,
by the theorem mentioned above, we can find two polynomials B
and C such that
and hence the preceding integral may be written in the form
f A_dx_ C B<}>+ T(V fBdx C $dx
J 4>* ~J ' V ~~J «#>"-1+J r f '
If n is greater than unity, taking
u=C, v = —
and integrating by parts, we get
Cc4Sdx= C 1 C C'
J 4? (n-\}r~l n-lj fi-»"
whence, substituting in the preceding equation, we find the formula
C A dx C C A^dx
J p '' ~(W-1)^"-1+J -&=*'
V, §104] RATIONAL FUNCTIONS 213
where A! is a new polynomial. If n > 2, we may apply the same
process to the new integral, and so on : the process may always be
continued until the exponent of <£ in the denominator is equal to
one, and we shall then have an expression of the form
A dx C \Ldx
*(*>+J V
where R(x) is a rational function of x, and ^ is a polynomial whose
degree we may always suppose to be less than that of <f>, but which
is not necessarily prime to <£. To integrate the latter form we must
know the roots of <£, but the evaluation of this integral will intro
duce no new rational terms, for the decomposition of the fraction
\[//<f> leads only to terms of the two types
A Mx + N
x-a (z-a)2 + /32'
each of which has an integral which is a transcendental function.
This method enables us, in particular, to determine whether the
integral of a given rational function is itself a rational function.
The necessary and sufficient condition that this should be true is
that each of the polynomials like ^ should vanish when the process
has been carried out as far as possible.
It will be noticed that the method used in obtaining the reduction formula
for /„ is essentially only a special case of the preceding method. Let us now
consider the more general integral
(Ax* + 2Bx + C)»
From the identity
A(Ax* + 2Bx + C)- (Ax + B)2 = AC- B*
it is evident that we may write
dx
C ** = A C
J (Ax* + 2Bx + C)« A C - B* J (Ax*
-S»J
(Ax + B)
2Bx + C)"-1
(Ax + B)dx
AC-B*J ^ ' (Ax* + 2Bx + C)«
Integrating the last integral by parts, we find
Ax + B Ax + B
(Ax -f B) dx = —
\ ' / A — O i C\ Tt — . y^v ..
(Ax* + 2Bx + C)n 2(n-
2n-2J (Ax* + 2Bx + C)n~
214 INDEFINITE INTEGRALS [V, § 104
whence the preceding relation becomes
Ax + B
(Ax* + 2Bx + C)» 2(n- \)(AC - B*)(Ax* + 2Bx
2n-3
^ /*
- £2 J (4z* + 2
2n - 2 AC - £2 J (Axi + 2£X + C]
Continuing the same process, we are led eventually to the integral
dx
^Ix2 + 2Bx + C'
which is a logarithm if B2 - AC>0, and an arctangent if B2 - AC<0.
As another example, consider the integral
C 5x3 + 3x - 1 .
dx.
J (x» + 3x+ ,
From the identity
5x3 + 3x - 1 = 6x(x2 + 1) - (x3 + 3x + 1)
it is evident that we may write
/5x8 + 3x — 1 C 6x(x2 + l) C
(x« + 3x + l)3dx=J ^^3x + l)*dX-J (xM
dx
+ OX +
Integrating the first integral on the right by parts, we find
dx
Cx 6(g8 + i)tfg = -x r
J (X3 + 3X + I)3 ~ (X3 + 3X + 1)2 + J (
(x8 + 3x + 1)2
whence the value of the given integral is seen to be
5x8 + 3x - 1 . - z
— -dx =
Note. In applying Hermite's method it becomes necessary to solve the fol
lowing problem : given three polynomials A, B, C, of degrees m, n, p, respectively,
two of which, A and B, are prime to each other, find two other polynomials u and v
such that the relation Au + Bv = C is identically satisfied.
In order to determine two polynomials u and v of the least possible degree
which solve the problem, let us first suppose that p is at most equal to m + n- 1.
Then we may take for u and v two polynomials of degrees n - I and m - 1,
respectively. The m + n unknown coefficients are then given by the system of
m + n linear non-homogeneous equations found by equating the coefficients.
For the determinant of these equations cannot vanish, since, if it did, we could
find two polynomials u and v of degrees n - 1 and m - 1 or less which satisfy
the identity Au + Bv = 0, and this can be true only when A and B have a
common factor.
If the degree of C is equal to or greater than m + n, we may divide C by AB
and obtain a remainder C" whose degree is less than m + n. Then C = A BQ + C',
and, making the substitution u - BQ = MI, the relation Au + Bv = C reduces to
Aui + Bv — C'. This is a problem under the first case.
V, §105] RATIONAL FUNCTIONS 215
105. Integrals of the type /R(x, ^Ax? + 2Bx + c) dx. After the
integrals of rational functions it is natural to consider the inte
grals of irrational functions. We shall commence with the case in
which the integrand is a rational function of x and the square root
of a polynomial of the second degree. In this case a simple substitu
tion eliminates the radical and reduces the integral to the preceding
case. This substitution is self-evident in case the expression under
the radical is of the first degree, say ax + b. If we set ax + b = t*,
the integral becomes
Cll(x, ^ax^b]dx = /V— , t] — >
J J \ a J a
and the integrand of the transformed integral is a rational function.
If the expression under the radical is of the second degree and
has two real roots a and b, we may write
A(x-a)(x-b) = (x-b)
and the substitution
Aa - bt2
or x = - — >
A - t*
actually removes the radical.
If the expression under the radical sign has imaginary roots, the
above process would introduce imaginaries. In order to get to the
bottom of the matter, let y denote the radical ^Ax2 -f 2Bx + C.
Then x and y are the coordinates of a point of the curve whose
equation is
(1) y2 = Ax2 + 2Bx + C,
and it is evident that the whole problem amounts to expressing the
coordinates of a point upon a conic by means of rational functions
of a parameter. It can be seen geometrically that this is possible.
For, if a secant
y - J3 = t(x - a)
be drawn through any point (a, /3) on the conic, the coordinates of
the second point of intersection of the secant with the conic are
given by equations of the first degree, and are therefore rational
functions of t.
If the trinomial Ax2 + 2Bx + C has imaginary roots, the coeffi
cient A must be positive, for if it is not, the trinomial will be
negative for all real values of x. In this case the conic (1) is an
216 INDEFINITE INTEGRALS [V, § 105
hyperbola. A straight line parallel to one of the asymptotes of
this hyperbola,
y = x Vyi + t,
cuts the hyperbola in a point whose coordinates are
C - t* r~ C - t*
2B 2t A - 2B
If A < 0, the conic is an ellipse, and the trinomial A x2 + 2Bx + C
must have two real roots a and b, or else the trinomial is negative
for all real values of x. The change of variable given above is pre
cisely that which we should obtain by cutting this conic by the
moving secant
y = t(x — a) .
As an example let us take the integral
(x2 + k) Vz2 + k
The auxiliary conic if = x2 + k is an hyperbola, and the straight line
x + y = t, which is parallel to one of the asymptotes, cuts the hyper
bola in a point whose coordinates are
Making the substitution indicated by these equations, we find
_ dt ft* + k\ C dx 4tdt 2
= '~
C dx _
J 7~
*•
or, returning to the variable x,
dx _ x — Vcc2 + k _ x 1
where the right-hand side is determined save for a constant term
In general, if A C — B'2 is not zero, we have the formula
Ix 1 Ax + B
(Ax2 + 2Bx + Cy AC — B2 VAx2 + 2Bx + C
In some cases it is easier to evaluate the integral directly without
removing the radical. Consider, for example, the integral
dx
+ 2Bx + C
V, §105] RATIONAL FUNCTIONS 217
ff the coefficient A is positive, the integral may be written
^/Adx C VJdx
r ^/Adx _ C
J ^A*x2 + 2ABx + AC J
x + £)2 + AC - B*
or setting Ax + B = t,
dt
i r
^AJ -
t-AC — B* -VA
Returning to the variable x, we have the formula
dx 1
f 2Bx + C
B +A Ax* + 2Bx
If the coefficient of x2 is negative, the integral may be written in
the form
/7 /*
doc I
V- Ax2 + 2Bx + C J VJ
B2 - (Ax - BY
The quantity A C + B2 is necessarily positive. Hence, making the
substitution
Ax - B = t ^/A C + B2,
the given integral becomes
dt I
_ r
I J
V.4 J Vl - t2 V.4
Hence the formula in this case is
dx 1 Ax — B
= —-^= arc sin
V- Ax2 + 2Bx + C
It is easy to show that the argument of the arcsine varies from — 1
to + 1 as x varies between the two roots of the trinomial.
In the intermediate case when ^4=0 and B =£ 0, the integral is
algebraic :
f
J
Integrals of the type
dx
(x - a) V^x2 + 2Bx + C
218
INDEFINITE INTEGRALS
[V, §100
reduce to the preceding type by means of the substitution x = a -f 1/y.
We find, in fact, the formula
r dx r
J (x — a) -^/Ax* + 2Bx + C J
dy
where
A ! = Aa2 + 2Ba + C, Bl = Aa + B,
It should be noticed that this integral is algebraic if and only if
the quantity a is a root of the trinomial under the radical.
Let us now consider the integrals of the type f Va:2 + A dx. Inte
grating by parts, we find
rVa;" + A
+ A dx = x Va:2 + A
On the other hand we have
fx2dx C I—. r Adx
. = I Va;2 -f A dx — I — — -
Va:2 + A J J Va;2 + J
= / Va:2 -\-Adx-A log (x + Va:2 + A ) .
From these two relations it is easy to obtain the formulae
c2 + A + - log (or + Va:2 + ^),
(2)
C
/
J
7 2
+ A *
The following formulae may be derived in like manner:
(5)
x*dx
— arc sin — •
z a
106. Area of the hyperbola. The preceding integrals occur in the evaluation
of the area of a sector of an ellipse or an hyperbola. Let us consider, for
example, the hyperbola
V, § 106]
RATIONAL FUNCTIONS
219
and let us try to find the area of a segment AMP bounded by the arc AM, the
x axis, and the ordinate MP. This area is equal to the definite integral
•'6
that is, by the formula (2),
- Vx2 — a2 dx ,
a
a2 - a2 log (?-±
But MP = y = (6/a) Vx2 — a2, and the term (b/2a) x Vx2 — a'2 is precisely the
area of the triangle OMP. Hence the area S of the sector 0AM, bounded by
the arc AM and the radii vectores OA
and OM, is
S = 1 e* loj
2
/x + Vx2 - a2\
I I
V a )
1 . , x
= - oft log ( -
2 \a
This formula enables us to express
the coordinates x and y of a, point M
of the hyperbola in terms of the area S.
In fact, from the above and from the
equation of the hyperbola, it is easy to
show that
FIG. 21
h
y = - (e"b — e
The functions which occur on the right-hand side are called the hyperbolic
cosine and sine :
e* + e~x
cosh x =
sinh x =
2 2
The above equations may therefore be written in the form
2S
x = a cosh
ab
, . .
y = b sinh —
ab
These hyperbolic functions possess properties analogous to those of the trigo
nometric functions.* It is easy to deduce, for instance, the following formulae :
cosh2x — sinh2x = 1,
cosh (x + y) = cosh x cosh y + sinh x sinh y,
sinh (x + y) = sinh x cosh y + sinh y cosh x.
* A table of the logarithms of these functions for positive values of the argument
is to be found in HoueTs Recueil desformules numeriques.
220
INDEFINITE INTEGRALS
[V, § 107
It may be shown in like manner that the coordinates of a point on an ellipse
may be expressed in terms of the area of the corresponding sector, as follows :
2S
x = a cos — ,
ab
, . 2S
y = b sin —
ao
In the case of a circle of unit radius, and in the case of an equilateral hyperbola
whose semiaxis is one, these formulae become, respectively,
x = cos2S, 7/ = sin2-S;
z = cosh2S, y = sinh2/S.
It is evident that the hyperbolic functions bear the same relations to the equi
lateral hyperbola as do the trigonometric functions to the circle.
107. Rectification of the parabola. Let us try to find the length of the arc of
a parabola 2py = x2 between the vertex 0 and any point M. The general
formula gives
. pj7w^ydz= r^sv
Jo * W J0 P
or, applying the formula (2),
2p 2 - \ p
The algebraic term in this result is precisely the length M T of the tangent,
for we know that OT = x/2, and hence
x2 x* x2 x2(x2 -4- v^)
4 4p* 4 4p2
If we draw the straight line connecting T to the focus F, the angle MTF will
be a right angle. Hence we
have
FT
Vn2 a-2 1
£H.J=I
/' whence we may deduce a curi
ous property of the parabola.
Suppose that the parabola
rolls without slipping on the x
axis, and let us try to find the
— i , " 7"" locus of the focus, which is sup
posed rigidly connected to the
parabola. When the parabola
is tangent at M' to the x axis, OM' = arc OM. The point T has come into a
position T' such that M'T' = 3/T, and the focus F is at a point F' which is
found by laying off T'F' = TF on a line parallel to the y axis. The coordi
nates X and Y of the point F' are then
T T'
FIG. 22
V,§108] RATIONAL FUNCTIONS 221
and the equation of the locus is given by eliminating x between these two equa
tions. From the first we find
!£
x + Vx2 + p2 = pe P ,
to which we may add the equation
_ _
x — Vx2 + j>2 = — pe P '
since the product of the two left-hand sides is equal to — p*. Subtracting these
two equations, we find
and the desired equation of the locus is
2 p
This curve, which is called the catenary, is quite easy to construct. Its form
is somewhat similar to that of the parabola.
108. Unicursal curves. Let us now consider, in general, the inte
grals of algebraic functions. Let
(6) F(x, y) = 0
be the equation of an algebraic curve, and let R(x, y) be a rational
function of x and y. If we suppose y replaced by one of the roots
of the equation (6) in R(x, y), the result is a function of the single
variable x, and the integral
is called an Abelian integral with respect to the curve (6). When
the given curve and the function R(x, y) are arbitrary these inte
grals are transcendental functions. But in the particular case where
the curve is unicursal, i.e. when the coordinates of a point on the
curve can be expressed as rational functions of a variable param
eter t, the Abelian integrals attached to the curve can be reduced at
once to integrals of rational functions. For, let
be the equations of the curve in terms of the parameter t. Taking
t as the new independent variable, the integral becomes
j
R(x, y}dx =
and the new integrand is evidently rational.
222 INDEFINITE INTEGRALS [v, § 108
It is shown in treatises on Analytic Geometry* that every uni-
cursal curve of degree n has (n — l)(n — 2)/2 double points, and,
conversely, that every curve of degree n which has this number of
double points is unicursal. I shall merely recall the process for
obtaining the expressions for the coordinates in terms of the param
eter. Given a curve CB of degree n, which has 8 = (n — !)(«. — 2)/2
double points, let us pass a one-parameter family of curves of degree
n — 2 through these 8 double points and through n — 3 ordinary points
on Ca. These points actually determine such a family, for
- 1,
whereas (n — 2)(n -(-l)/2 points are necessary to determine uniquely
a curve of order n — 2. Let P(x, y) + tQ(x, ?/) = 0 be the equation
of this family, where t is an arbitrary parameter. Each curve of the
family meets the curve Cn in n(n — 2) points, of which a certain num
ber are independent of t, namely the n — 3 ordinary points chosen
above and the 8 double points, each of which counts as two points of
intersection. But we have
» - 3 + 28 = n - 3 + (ft -l)(n - 2) = n(n - 2) -1,
and there remains just one point of intersection which varies with t.
The coordinates of this point are the solutions of certain linear equa
tions whose coefficients are integral polynomials in t, and hence they
are themselves rational functions of t. Instead of the preceding we
might have employed a family of curves of degree n — 1 through the
(n — l)(w — 2)/2 double points and 2n — 3 ordinary points chosen at
pleasure on Cn.
If n = 2, (n — l)(w — 2)/2 = 0, — every curve of the second
degree is therefore unicursal, as we have seen above. If n = 3,
(n — l)(w — 2)/2 = 1, — the unicursal curves of the third degree
are those which have one double point. Taking the double point
as origin, the equation of the cubic is of the form
4>s (x, y) + fa (x, y) = 0 ,
where <£3 and <£2 are homogeneous polynomials of the degree of their
indices. A secant y = tx through the double point meets the cubic
in a single variable point whose coordinates are
«£«(!, Q <&(!, Q
<MM)' " *i(M)'
*See, e.g., Niewenglowski, Cours de Geometric analytique, Vol. II, pp. 99-114.
V, §108] .RATIONAL FUNCTIONS 223
A unicursal curve of the fourth degree has three double points.
In order to find the coordinates of a point on it, we should pass a
family of conies through the three double points and through another
point chosen at pleasure on the curve. Every conic of this family
would meet the quartic in just one point which varies with the
parameter. The equation which gives the abscissae of the points of
intersection, for instance, would reduce to an equation of the first
degree when the factors corresponding to the double points had
been removed, and would give x as a rational function of the
parameter. We should proceed to find y in a similar manner.
As an example let us consider the lemniscate
which has a double point at the origin and two others at the imagi
nary circular points. A circle through the origin tangent to one of
the branches of the lemniscate,
x* + y2 = t(x - y} ,
meets the curve in a single variable point. Combining these two
equations, we find
or, dividing by x — y,
This last equation represents a straight line through the origin which
cuts the circle in a point not the origin, whose coordinates are
_ oa*(<3 + a2) _ a2t(t2 - a2)
t* + a* t* + a<
These results may be obtained more easily by the following
process, which is at once applicable to any unicursal curve of the
fourth degree one of whose double points is known. The secant
y = \x cuts 'the lemniscate in two points whose coordinates are
The expression under the radical is of the second degree. Hence,
by § 105, the substitution (1 - X)/(l + A) = (a/t)'2 removes the radi
cal. It is easy to show that this substitution leads to the expressions
just found.
224 INDEFINITE INTEGRALS [V, § 109
Note. When a plane curve has singular points of higher order, it
can be shown that each of them is equivalent to a certain number of
isolated double points. In order that a curve be unicursal, it is suffi
cient that its singular points should be equivalent to (n — l)(n — 2)/2
isolated double points. For example, a curve of order n which has
a multiple point of order n — 1 is unicursal, for a secant through
the multiple point meets the curve in only one variable point.
109. Integrals of binomial differentials. Among the other integrals
in which the radicals can be removed may be mentioned the follow
ing types :
/ R\_x, (ax + b)'1 \dx , I R(X, ~vax -f- b, ~V 'ex + d)dx,
R(xa, xa\ xa", • • -)dx,
where R denotes a rational function and where the exponents
a, a', a", ••• are commensurable numbers. For the first type it is
sufficient to set ax -f b = tq. In the second type the substitution
ax + b = t* leaves merely a square root of an expression of the
second degree, which can then be removed by a second substitution.
Finally, in the third type we may set x = tD, where D is a common
denominator of the fractions n-, a', a",
In connection with the third type we may consider a class of
differentials of the form
which are called binomial differentials. Let us suppose that the
three exponents m, n, p are commensurable. If p is an integer, the
expression may be made rational by means of the substitution
x = tD, as we have just seen. In order to discover further cases
of integrability, let us try the substitution axn + b — t. This gives
\ x™ (axn + by dx = — |W^— -H
J naj \ a /
dx — —
\ a /
" dt.
The transformed integral is of the same form as the original, and
the exponent which takes the place of p is (m + 1) /n — 1. Hence
the integration can be performed if (m + l)/w is an integer.
V,§109] RATIONAL FUNCTIONS 225
On the other hand, the integral may be written in the form
whence it is clear that another case of integrability is that in which
(m + np 4- l)/w = (m + l)/w + p is an integer. To sum up, the
integration can be performed whenever one of the three numbers
2^ (in +l)/n, (m+V)/n +p is an integer. In no other case can the
integral be expressed by means of a finite number of elementary
functional symbols when m, n, and p are rational.
In these cases it is convenient to reduce the integral to a simpler
form in which only two exponents occur. Setting axn = bt, we find
/As l 1/Asl-,
x = (- } tn, dx = - - ) tn dt,
a n a
IP /*\!i±i r "1 + 1 i
xm(axn + bydx = - - " ltn
n \a/ J
Neglecting the constant factor and setting q = (m + l)/n — 1, we
are led to the integral
/•
tydt.
The cases of integrability are those in which one of the three num
bers p, q, p + q is an integer. If p is an integer and q = r/s, we
should set t — us. If q is an integer and p — r/s, we should set
l+t = u*. Finally, if p + q is an integer, the integral may be
written in the form
and the substitution 1 + t = tus, where p — r/s, removes the radical.
As an example consider the integral
x Vl + x3 dx .
Here m = 1, n = 3, p = 1/3, and (m + l)/w + p =• 1. Hence this
is an integrable case. Setting x3 = t, the integral becomes
— dt,
and a second substitution 1 + t = tu8 removes the radical.
226 INDEFINITE INTEGRALS [V, § 110
II. ELLIPTIC AND HYPERELLIPTIC INTEGRALS
110. Reduction of integrals. Let P(x) be an integral polynomial
of degree/? which is prime to its derivative. The integral
where R denotes a rational function of x and the radical y — Vp(ce),
cannot be expressed in terms of elementary functions, in general,
when p is greater than 2. Such integrals, which are particular
cases of general Abelian integrals, can be split up into portions which
result in algebraic and logarithmic functions and a certain number
of other integrals which give rise to new transcendental functions
which cannot be expressed by means of a finite number of elemen
tary functional symbols. We proceed to consider this reduction.
The rational function R(x, y) is the quotient of two integral
polynomials in x and y. Replacing any even power of y, such as
y2q> by [P(V)]9> and any odd power, such as y*g + l, by y [_P(x)~]q, we
may evidently suppose the numerator and denominator of this frac
tion to be of the first degree in y,
A + By
R(x, in — >
v ' J) C + Dy
where A, B, C, D are integral polynomials in x. Multiplying the
numerator and the denominator each by C — Dy, and replacing y2
by P(x), we may write this in the form
R(x, y} --
K
where F, G, and K are polynomials. The integral is now broken
up into two parts, of which the first JF/K dx is the integral of a
rational function. For this reason we shall consider only the second
integral fOy/K dx, which may also be written in the form
fpft
where M and N are integral polynomials in x. The rational frac
tion M/N may be decomposed into an integral part E(x) and a
sum of partial fractions
V,§lio] ELLIPTIC AND HYPERELLIPTIC INTEGRALS 227
where each of the polynomials A',- is prime to its derivative. We
shall therefore have to consider two types of integrals,
/x
V
ldx C Adx
If the degree of P(x) is p, all the integrals Ym may be expressed
in terms of the first p — 1 of them, F0, Yi} ••-, Yp_2, and certain
algebraic expressions.
For, let us write
P(X) = a0xp + a^x* + • • ••
It follows that
~JZ(A
2 VP(Z)
_ 2mxm-1P(x) + xmP'(x)
2
The numerator of this expression is of degree m + p — 1, and its
highest term is (2m + p~)a0xm+p~l. Integrating both sides of the
above equation, we find
2x'»Vp(xj=(2m+p}a0Ym+p_l + .--,
where the terms not written down contain integrals of the type
Y whose indices are less than m + p — 1. Setting m = 0, 1, 2, • • •,
successively, we can calculate the integrals Yp_l} Yp, ••• succes
sively in terms of algebraic expressions and the p — 1 integrals
YO> YI, • • •, Yp_2.
With respect to the integrals of the second type we shall distin
guish the two cases where X is or is not prime to P(x)>
1) If X is prime to P(x), the integral Zn reduces to the sum of
an algebraic term, a number of integrals of the type Yk, and a new
integral
B dx
f-
X V P(x)
where B is a polynomial whose degree is less than that of X.
Since X is prime to its derivative X ' and also to P(ar), Xn is prime
to PA". Hence two polynomials A and fi can be found such that
XX" + p.X'P = A, and the integral in question breaks up into
two parts:
f— ii== f *^_ i c
J Xn^P(x) J VP(X) J
Xn
228 INDEFINITE INTEGRALS [V, § 110
The first part is a sum of integrals of the type Y. In the second
integral, when n > 1, let us integrate by parts, taking
— — 1
f I ~\./ T-) — • *3/ tt TT~" • »
** (n -I)*-1
which gives
r^px'dx= -pV7> l i rwp + pp* ^
J Xn (n-l)Xn~l n-lJ 2J£"-1VP(a:)
The new integral obtained is of the same form as the first, except
that the exponent of X is diminished by one. Repeating this
process as often as possible, i.e. as long as the exponent of X is
greater than unity, we finally obtain a result of the form
'P(x) J .YVP J VP Xn~
where B, C, D are all polynomials, and where the degree of B may
always be supposed to be less than that of X.
2) If X and P have a common divisor D, we shall have X = YD,
P = SD, where the polynomials D, S, and Y are all prime to each
other. Hence two polynomials X and /u. may be found such that
A = XDn + p.Y", and the integral may be written in the form
/Adx C \dx C fidx
jpVp J rnVp J Dn
The first of the new integrals is of the type just considered. The
second integral,
where D is a factor of P, reduces to the sum of an algebraic term
and a number of integrals of the type Y.
For, since Dn is prime to the product D'S, we can find two poly
nomials A! and //,! such that X^" -f piD'S = /x. Hence we may write
C »dx_= C\±dx+ Cr^
J £>nVP J VP J D"
'VP
Replacing P by DS, let us write the second of these integrals in the
form
D'
V,§110] ELLIPTIC AND HYPERELLIPTIC INTEGRALS 229
and then integrate it by parts, taking
-11
V — - -
which gives
Cjidx^ r, |
J D"VP J
This is again a reduction formula ; but in this case, since the expo
nent n — 1/2 is fractional, the reduction may be performed even
when D occurs only to the first power in the denominator, and we
finally obtain an expression of the form
C p.dx _ KVP C
J Dn VP "fP V
Hdx
where H and K are polynomials.
To sum up our results, we see that the integral
M dx
can always be reduced to a sum of algebraic terms and a number of
integrals of the two types
/rpftl ///•)• / "V /7'Tf
JO \JiJCi -**-\ UJU
VP ' J xVp
where ra is less than or equal to p — 2, where X is prime to its
derivative X' and also to P, and where the degree of ^ is less than
that of X. This reduction involves only the operations of addition,
multiplication, and division of polynomials.
If the roots of the equation X = 0 are known, each of the rational
fractions Xl/X can be broken up into a sum of partial fractions of
the two forms
A Bx + C
x-a (x-af + p*
where A, B, and C are constants. This leads to the two new types
r dx C (Bx
J x - a VP(V>' J x - a)2
(x - a) VP(V> J [_(x - a)2 + /32
which reduce to a single type, namely the first of these, if we agree
to allow a to have imaginary values. Integrals of this sort are
230 INDEFINITE INTEGRALS [V, § 110
called integrals of the third kind. Integrals of the type Ym are
called integrals of the first kind when ra is less than p/2 — 1, and
are called integrals of the second kind when m is equal to or greater
than p/2, — 1. Integrals of the first kind have a characteristic
property, — they remain finite when the upper limit increases
indefinitely, and also when the upper limit is a root of P(x)
(§§ 89, 90); but the essential distinction between the integrals of
the second and third kinds must be accepted provisionally at this
time without proof. The real distinction between them will be
pointed out later.
Note. Up to the present we have made no assumption about the
degree p of the polynomial P(x). If p is an odd number, it may
always be increased by unity. For, suppose that P(x) is a poly
nomial of degree 2q — 1 :
P(x) = A0x*»-1 + A.x^-* + .-. + A.2q_v.
Then let us set x = a -f- 1/y, where a is not a root of P(z). This
gives
I p(2<,-l)(fl\ 1 P,((/\
P(x)=P(a) + P'(a) i + ... + I__Lii -i-j = fJ-134,
y (2q - 1)! y-'< 1 i/2"
where P^ (y) is a polynomial of degree 2q. Hence we have
and any integral of a rational function of x and Vp(a-) is trans
formed into an integral of a rational function of y and '\/Pl(y).
Conversely, if the degree of the polynomial P(x) under the radi
cal is an even number 2q, it may be reduced by unity provided a
root of P(x) is known. For, if a is a root of P(x), let us set
x = a -f- 1/y. This gives
y (2?)! y»« y
where Pi(y) is of degree 2y — 1, and we shall have
Hence the integrand of the transformed integral will contain no
other radical than
V, §lllj ELLIPTIC AND IIYPERELLIPTIC INTEGRALS 231
111. Case of integration in algebraic terms. We have just seen that an integral
of the form
C R[X, VP(x)]dx
can always be reduced by means of elementary operations to the sum of an inte
gral of a rational fraction, an algebraic expression of the form G VP(x)/L, and
a number of integrals of the first, second, and third kinds. Since we can also
find by elementary operations the rational part of the integral of a rational
fraction, it is evident that the given integral can always be reduced to the form
]dx = F[x, VP(xj] + T,
where F is a rational function of x and VP(x), and where T is a sum of inte
grals of the three kinds and an integral fXi /Xdx, X being prime to its deriva
tive and of higher degree than X\ . Liouville showed that if the given integral
is integrable in algebraic terms, it is equal to F[x, VP(x)]. We should there
fore have, identically,
R[x, VP(x)~\ = ~
and hence T = 0.
Hence we can discover by means of multiplications and divisions of polynomials
whether a given integral is integrable in algebraic terms or not, and in case it is,
the same process gives the value of the integral.
112. Elliptic integrals. If the polynomial P(x) is of the second
degree, the integration of a rational function of x and P(x) can be
reduced, by the general process just studied, to the calculation of the
integrals
/dx C dx
VP(z) ' J (x - a)VP(z)'
which we know how to evaluate directly (§ 105).
The next simplest case is that of elliptic integrals, for which P(x)
is of the third or fourth degree. Either of these cases can be
reduced to the other, as we have seen just above. Let P(x) be a
polynomial of the fourth degree whose coefficients are all real and
whose linear factors are all distinct. We proceed to show that
a real substitution can always be found which carries P(x~) into a
polynomial each of whose terms is of even degree.
Let a, b, c, d be the four roots of P(x). Then there exists an
involutory relation of the form
(7) Lx'x" + M(x' + x") + N = 0,
232 INDEFINITE INTEGRALS [V, §112
which is satisfied by x' = a, x" — b, and by x' = c, x" = d. For the
coefficients L, M, N need merely satisfy the two relations
Lab + M(a + b) -f N — 0,
Led + M(c + d) + N = 0,
which are evidently satisfied if we take
L = a -+- b — c — d, M = cd — ab , N = ab (c + d) — cd (a + b).
Let a and ft be the two double points of this involution, i.e. the
roots of the equation
Zt*2 + 2ATw + N = 0.
These roots will both be real if
(cd - ab)*-(a + b - c - d) [ai (c + d) - cd(a + J)] > 0,
that is, if
(8) (a-c)(a-d)(b-c)(b-d)>Q.
The roots of P(x) can always be arranged in such a way that this
condition is satisfied. If all four roots are real, we need merely
choose a and b as the two largest. Then each factor in (8) is positive.
If only two of the roots are real, we should choose a and b as the real
roots, and c and d as the two conjugate imaginary roots. Then the
two factors a — c and a — d are conjugate imaginary, and so are the
other two, b — c and b — d. Finally, if all four roots are imaginary,
we may take a and b as one pair and c and d as the other pair of
conjugate imaginary roots. In this case also the factors in (8) are
conjugate imaginary by pairs. It should also be noticed that these
methods of selection make the corresponding values of L, M, N real.
The equation (7) may now be written in the form
(9} ~- 4- x"~a = 0
x> _ ft + X" - (3
If we set (x — a)/(x — ft~) = y, or x = (ft// — <*)/(y — 1), we find
where P\(y) is a new polynomial of the fourth degree with real
coefficients whose roots are
a — a b — a c — a d — a
a-p' b- ft' c- ft d- ft'
It is evident from (9) that these four roots satisfy the equation
V, §112] ELLIPTIC AND HYPERELLIPTIC INTEGRALS 233
y' -f- y" = 0 by pairs ; hence the polynomial /'i(y) contains no term
of odd degree.
If the four roots «, b, c, d satisfy the equation a + b = c + d, we
shall have L = 0, and one of the double points of the involution lies
at infinity. Setting a— — N/2M, the equation (7) takes the form
x' — a + x" — a — 0,
and we need merely set x = a + y in order to obtain a polynomial
which contains no term of odd degree.
We may therefore suppose P(x) reduced to the canonical form
It follows that any elliptic integral, neglecting an algebraic term
and an integral of a rational function, may be reduced to the sum
of integrals of the forms
dx C xdx C
•^AtX*+AiX*+Ai J ^/A^+AiXt+At' J
x*dx
^/A^+AiXt+At' J •^A9x*+Al
and integrals of the form
/;
dx
(x — a
The integral
dx
is the elliptic integral of the first kind. If we consider x, on the
other hand, as a function of u, this inverse function is called an
elliptic function. The second of the above integrals reduces to an
elementary integral by means of the substitution x2 = u. The third
integral
x2dx
is Legendre's integral of the second kind. Finally, we have the
identity
/dx _ C xdx C dx
(x - a)Vp(x) ~J (x2 - a2)VP(x) V (x2 - a2) VP(^)
The integral
dx
(x2 + /i) VJ0x4 + /Ijx2 + A.4
is Legendre's integral of the third kind.
234 INDEFINITE INTEGRALS [V, §113
These elliptic integrals were so named because they were first
met with in the problem of rectifying the ellipse. Let
x = a cos <f> , y = b sin <f>
be the coordinates of a point of an ellipse. Then we shall have
ds2 = dx2 + dy* = (a2 sin2<£ + t>* cos2<£) d<j>2,
or, setting a2 — b3 = e2a2,
ds = a Vl — e2 cos2 <£ d<f> .
Hence the integral which gives an arc of the ellipse, after the sub
stitution cos <£ = t, takes the form
It follows that the arc of an ellipse is equal to the sum of an inte
gral of the first kind and an integral of the second kind.
Again, consider the lemniscate defined by the equations
y = a"
.
t* + a4 t4 + a4
An easy calculation gives the element of length in the form
a
ds2 = dx2 + dif = - - -
t ~p Ct
dt*.
Hence the arc of the lemniscate is given by an elliptic integral of
the first kind.*
113. Pseudo-elliptic integrals. It sometimes happens that an integral of the
form f F[x, VP(x)] dx, where P(x) is a polynomial of the third or fourth
degree, can be expressed in terms of algebraic functions and a sum of a finite
number of logarithms of algebraic functions. Such integrals are called pseudo-
elliptic. This happens in the following general case. Let
(10) Lx'x" + M(x' + x") + N=0
be an involutory relation which establishes a correspondence between two pairs of
the four roots of the quartic equation P(x) = 0. If the function f(x) be such that
the relation
(ID
Lx
is identically satisfied, the integral /[/(x)/VP(x)] dx is pseudo-elliptic.
* This is a common property of a whole class of curves discovered by Serret
(Cours de Calcul differentiel et integral, Vol. II, p. 264).
V, §113] ELLIPTIC AND HYPERELLIPTIC INTEGRALS 235
Let a and p be the double points of the involution. As we have already
seen, the equation (10) may be written in the form
(12)
x' - /3 x" - p
Let us now make the substitution (z — a)/(x — P) = y. This gives
(a-^, P(2) = ^L,
(l-y)a (1-y)*
and consequently
dx _ (« — p) dy
where PI (y) is a polynomial of the fourth degree which contains no odd powers
of y (§ 112). On the other hand, the rational fraction f(x) goes over into a
rational fraction 4>(y), which satisfies the identity <p(y) + <£( — y) = 0. For if
two values of x correspond by means of (12), they are transformed into two
values of y, say y' and y" ', which satisfy the equation y' + y" = 0. It is evident
that 4>(y) is of the form y^(yz), where ^ is a rational function of y2. Hence
the integral under discussion takes the form
and we need merely set y2 = z in order to reduce it to an elementary integral.
Thus the proposition is proved, and it merely remains actually to carry out
the reduction.
The theorem remains true when the polynomial P(x) is of the third degree,
provided that we think of one of its roots as infinite. The demonstration is
exactly similar to the preceding.
If, for example, the equation P(x) = 0 is a reciprocal equation, one of the
involutory relations which interchanges the roots by pairs is x'x" = 1. Hence,
if f(x) be a rational function which satisfies the relation /(x) + /(1/x) = 0,
the integral /[/(»)/ VP(x)] dx is pseudo-elliptic, and the two substitutions
(x — l)/(z + 1) = y, y2 = z, performed in order, transform it into an elementary
integral.
Again, suppose that P(x) is a polynomial of the third degree,
Let us set a = o>, b = 0, c = 1, d = I/A;2. There exist three involutory rela
tions which interchange these roots by pairs :
l-Jk2x"
Hence, if /(x) be a rational function which satisfies one of the identities
236 INDEFINITE INTEGRALS [V, §114
the integral
f(z)dx
Vx(l- «)(!
is pseudo-elliptic. From this others may be derived. For instance, if we set
z = 22, the preceding integral becomes
whence it follows that this new integral is also pseudo-elliptic if /(z2) satisfies
one of the identities
The first of these cases was noticed by Euler.*
III. INTEGRATION OF TRANSCENDENTAL FUNCTIONS
114. Integration of rational functions of sin x and cos x. It is well
known that since and cos a? may be expressed rationally in terms
of tan ce/2 = t. Hence this change of variable reduces an integral
of the form
sinx, cosxjdx
to the integral of a rational function of t. For we have
2dt 2t l-t2
x = 2 arc tan t , ax = - - -^ > sin x = 2 > cos x = >
and the given integral becomes
where &(t) is a rational function. For example,
C dx Cdt
\ - - = I — = log t ;
J sin a; J t
hence
/dx
sin a;
dx x
= log tan--
sin a; 2
* See Hermite'3 lithographed Cours, 4th ed., pp. 25-28.
V, §114] TRANSCENDENTAL FUNCTIONS 237
The integral f [I/cos x~]dx reduces to the preceding by means of the
substitution x = 7r/2 — y, which gives
/dx
cos x
dx ITT x\ ITT x
- = - log tan T — - I = log tan - + -
coscc \4 2/ \4 2
The preceding method has the advantage of generality, but it is
often possible to find a simpler substitution which is equally suc
cessful. Thus, if the function /(sin x, cos x~) has the period IT, it is
a rational function of tana;, F(tan x). The substitution t&nx = t
therefore reduces the integral to the form
f
F(tan x~) dx =
As an example let us consider the integral
dx
A cos2 x -f- B sin x cos x + C sin2 x + D
where A, B, C, D are any constants. The integrand evidently has the
period IT ; and, setting tan x = t, we find
1
cos"1* x = ) sin x cos x — > sin2 x =
-L ~~T~ t 1 ~T~ *
Hence the given integral becomes
r
J
The form of the result will depend upon the nature of the roots
of the denominator. Taking certain three of the coefficients zero,
we find the formulae
/dx r dx
— — = tan:r, / -; - = log tan x,
cos2 x J sin x cos x
/;
dx
r-r- = — COt X.
When the integrand is of the form R(sm x) cos x, or of the form
R (cos x) since, the proper change of variable is apparent. In the
first case we should set sin x = t ; in the second case, cos x = t.
It is sometimes advantageous to make a first substitution in order
to simplify the integral before proceeding with the general method.
For example, let us consider the integral
dx
a cos x + b sin x -f
238 INDEFINITE INTEGRALS [V, § 114
where a, b, c are any three constants. If p is a positive number
and <£ an angle determined by the equations
a = p cos </>, b = p sin <f>,
we shall have
/ a , n a • b
p = V a -f- 0 ) cos o> = — 7== > sin d> = — p== >
Va2 + b2 Va2 + 62
and the given integral may be written in the form
/dx _ C dy
p cos (x -</>)+ c J p cos y + c'
where x — <f> = y. Let us now apply the general method, setting
tan y/2 — t. Then the integral becomes
2dt
and the rest of the calculation presents no difficulty. Two different
forms will be found for the result, according as p2 — c2 = a2 + b2 — c2
is positive or negative.
The integral
" m cos x -f n sin x + p .
; — : *- dx
a cos x + o sin x + c
f
may be reduced to the preceding. For, let u = a cos x + b sin x + c,
and let us determine three constants X, p., and v such that the equation
du
m cos x -\- n sin x + 7> = MI + u. — — \- v
ax
is identically satisfied. The equations which determine these num
bers are
m = \a + p.b, n = Xb — p,a, p = \c + v,
the first two of which determine X and /A. The three constants hav
ing been selected in this way, the given integral may be written in
the form
/du
Xt< + fi — + v
'. dx = \x -\- u. log u 4- v / —
u J a
cos x + b sin x + c
Example. Let us try to evaluate the definite integral
dx
1 + ecosx
where |e|<l.
y, §115] TRANSCENDENTAL FUNCTIONS 239
Considering it first as an indefinite integral, we find successively
C dx _2 C dt 2 C du
J 1 + ecosx J l + e + (l- e)V ~ vT^J 1 + w2'
by means of the successive substitutions tanx/2 = f, t = u V(l + e)/(l — e).
Hence the indefinite integral is equal to
2 / ll-e. x\
. arc tan I \l tan - J •
TIT^ \ \ i + c 2/
As x varies from 0 to x, V(l — e)/(l + e) tan x/2 increases from 0 to + oo, and
the arctangent varies from 0 to it/2. Hence the given definite integral is equal
to ;r/V(l - e2).
115. Reduction formulae. There are also certain classes of integrals
for which reduction formulae exist. For instance, the formula for
the derivative of tann~l£ may be written
j- (tan"-1a;) = (?i -l)tan"-2x(l+ tan2a;),
whence we find
/tan" ~ l x C
ka,n"xdx = — -- / tan"~
n-l J
The exponent of tan x in the integrand is diminished by two units.
Repeated applications of this formula lead to one or the other of
the two integrals
I dx = x , I tan x dx = — log cos x .
The analogous formula for integrals of the type /cot" x dx is
C cot"-1^ r
I c*r\i~ ** T* fiw — — - - ____ __ I r»/-kir'* — * '»* fit*
I L/UU «<C W-i-C — ^^ I CUu X (L3C ,
J n-l J
In general, consider the integral
sinmx cosnxdx,
where m and n are any positive or negative integers. When one of
these integers is odd it is best to use the change of variable given
above. If, for instance, n = 2p + 1, we should set sin x = t, which
reduces the integral to the form />(! — t2)p dt.
Let us, therefore, restrict ourselves to the case where m and n are
both even, that is, to integrals of the type
= fsi
J
sin2'" a; cosj"ccc?x,
240 INDEFINITE INTEGRALS [V, § 116
which may be written in the form
T = I sin2m~lxcos'2nxsmxdx.
*m,m J
Taking cos2nz since dx as the differential of [— l/(2n -f I)]cos2n + 1a;,
an integration by parts gives
pQgZn-fl^ 2m 1 C
T = -sin2m-lx- r + - — — r / sinsm-8ajcos*"aj(l-sinax)cte,
-Lm'n 2ft + 1 2n+lJ
which may be written in the form
7" sin2"*"1^ cos2"+1a; 2m — 1 j
-*•*• ~ 2 (m + n) 2 (m + ri) -*»-!,•'
This formula enables us to diminish the exponent m without alter
ing the second exponent. If m is negative, an analogous formula
may be obtained by solving the equation (A) with respect to /m_lin
and replacing m by 1 — m :
j sin1~2?'a; cos2n+1a; 2 (n — m + 1) ,-
•*—*•.. 1-2OT 1-2WI ^l-m,n'
The following analogous formulae, which are easily derived, enable
us to reduce the exponent of cos a; :
r sin2m+1a;cos2n~1a; 2n — 1 T
•••M : 2 (m + w) 2(m + w) ^m,«-i'
^- _ sin»m+1a;cos1~8*a; 2 (m + 1 — n) j
( t **,-• l—2n l—2n -*«,-•+!*
Eepeated applications of these formulae reduce each of the num
bers m and n to zero. The only case in which we should be unable to
proceed is that in which we obtain an integral /TOiB, where m + n = 0.
But such an integral is of one of the types for which reduction for
mulae were derived at the beginning of this article.
116. Wallis' formulae. There exist reduction formulae whether the exponents
m and n are even or odd.
As an example let us try to evaluate the definite integral
•n
Im = I s'mmxdx,
Jo
where m is a positive integer. An integration by parts gives
7T 7T E
| 2sium-1xsinxtZx = — [cosxsin™-^]^ + (m — 1) | 2 sinm-2cos2xdx,
Jo o Jo
V, §117] TRANSCENDENTAL FUNCTIONS 241
whence, noting that cosz s\nm~lx vanishes at both limits, we find the formula
7T
Im = (m - 1) f 2sin»»-2x(l - sin2x)<Jx = (m - l)(/m_2 - /m),
Jo
ivhich leads to the recurrent formula
m — 1 T
(13) Im = - - Im-2-
m
Repeated applications of this formula reduce the given integral to IQ = it/2
if m is even, or to Ii = 1 if m is odd. In the former case, taking m = 2p and
replacing m successively by 2, 4, 6, • • • , 2j>, we find
1 T T 3 T T - 2P-1 T
-*a = --'05 -M = 7 -*2> '*« J2p— — 2n~
or, multiplying these equations together,
_ 1 . 3 . 6 • • • (2p - 1) 5
2 . 4 . 6 • • • 2p 2 '
Similarly, we find the formula
2 . 4 . 6 • • • 2p
12.P + 1 =
1 . 3 . 5 • • • (2j> + 1)
A curious result due to Wallis may be deduced from these formulae. It is
evident that the value of Im diminishes as m increases, for sinm+1x is less than
sinmz. Hence
and if we replace I2p + i> ^2P, ISP-I by their values from the formulae above, we
find the new inequalities
where we have set, for brevity,
2244 2p - 2 2p
1335 2p - 1 2p - 1
It is evident that the ratio n/2Hp approaches the limit one as p increases indefi
nitely. It follows that 7T/2 is the limit of the product Hp as the number of
factors increases indefinitely. The law of formation of the successive factors is
apparent.
117. The integral /cos (ax + b) cos (a'x + b') • • • dx. Let us consider
a product of any number of factors of the form cos (ax + b), where
a and b are constants, and where the same factor may occur several
times. The formula
cos (u + ?') cos (u — v)
cos u cos v = ^ H ^
242 INDEFINITE INTEGRALS [V, § 117
enables us to replace the product of two factors of this sort by the
sum of two cosines of linear functions of x ; hence also the product
of n factors by the sum of two products of n — 1 factors each.
Kepeated applications of this formula finally reduce the given inte
gral to a sum of the form 2 H cos (Ax + B), each term of which is
immediately integrable. If A is not zero, we have
/ , sin (Ax 4- B}
cos (Ax -f B}dx = » — — I + C,
J\.
while, in the particular case when A = 0, /cos B dx = x cos B + C.
This transformation applies in the special case of products of
the form
cosma; sinnic,
where m and n are both positive integers. For this product may
be written
and, applying the preceding process, we are led to a sum of sines and
cosines of multiples of the angle, each term of which is immediately
integrable.
As an example let us try to calculate the area of the curve
which we may suppose given in the parametric form x = acos'0,
y = b sin'0, where 6 varies from 0 to 2-rr for the whole curve. The
formula for the area of a closed curve,
A = - / xdy — ydx,
i/(C)
gives
I
Jo
But we have the formula
(sin 6 cos 0)2 = - sin220 = - (1 - cos
Hence the area of the given curve is
Sab [~ sin40~l27r 3-rrab
V, §117]
TRANSCENDENTAL FUNCTIONS
243
It is now easy to deduce the following formulae •.
/I — cos 2x
a; sin 2x
hr<
sin2x dx
2
/3 sin x — sin 3x ^
2 4
3 cos cc
c>
cos 3cc j
1 sin3 a? dx
r .
4
/3 — 4 cos 2x 4- cos 4x ^
3ic sin 2x
12 '
, sin 4x
| sin4cc dx
8
S 4
32
r
T1+OOS2X
a; sin 2x
1 cos x dx
J 2
/3 cos cc + cos 3x j
2 ' 4
3 sin x t si
T t7>
n3x
1 cos3 a; dx
r 4
4
/3 + 4 cos 2x 4- cos 4x ^
4
3x t sin2x
12 C'
J °
8 4
32
C,
A general law may be noticed in these formulae. The integrals
F(x) — f*sin.nxdx and <J>(cc) = f* cos"x dx have the period 2?r
when n is odd. On the other hand, when n is even, these integrals
increase by a positive constant when x increases by 2?r. It is evi
dent a priori that these statements hold in general. For we have
F(x + 2-Tr)
/-»2Jr r»27r + a:
= I sin";rcfa; + I sin"
Jo Jz-rr
xdx,
or
/o2ir r>x /^2ir
x + 27r) = I sinna;c?ic+ I sinna;rfa; = F(x~) + I sinn£co?a;,
i/O i/O i/O
F(
since sin x has the period 2?r. If n is even, it is evident that the
integral f2vsinnx dx is a positive quantity. If n is odd, the same
integral vanishes, since sin (x + TT) = — sin x.
Note. On account of the great variety of transformations appli
cable to trigonometric functions it is often convenient to introduce
them in the calculation of other integrals. Consider, for example,
the integral /[!/(!+ x*)*~\dx. Setting x = ts,n<f>, this integral
becomes f cos </> d<j> = sin <j> + C. Hence, returning to the variable x,
dx
which is the result already found in § 105.
244 INDEFINITE INTEGRALS [V, § 118
118. The integral /R(x)ewxdx. Let us now consider an integral
of the form /R(x)e"xdx, where R (x) is a rational function of x.
Let us suppose the function R(x) broken up, as we have done
several times, into a sum of the form
where E(x), Al, A2, • • •, Ap, X1} • • •, Xp are polynomials, and Xt is
prime to its derivative. The given integral is then equal to the
sum of the integral / E(x)e'axdx, which we learned to integrate in
§ 85 by a suite of integrations by parts, and a number of integrals
of the form
There exists a reduction formula for the case when n is greater
than unity. For, since X is prime to its derivative, we can determine
two polynomials A. and /* which satisfy the identity A = \X + p.X'.
Hence we have
and an integration by parts gives the formula
— ; dx.
f
J
-1 X"
Uniting these two formulas, the integral under consideration is
reduced to an integral of the same type, where the exponent n is
reduced by unity. Eepeated applications of this process lead to
the integral
dx,
X
where the polynomial B may always be supposed to be prime to
and of less degree than X. The reduction formula cannot be applied
to this integral, but if the roots of X be known, it can always be
reduced to a single new type of transcendental function. For
defmiteness suppose that all the roots are real. Then the integral
in question can be broken up into several integrals of the form
— dx.
x — a
v, §119] TRANSCENDENTAL FUNCTIONS 245
Neglecting a constant factor, the substitutions x = a + y/u>, u = e"
enable us to write this integral in either of the following forms :
/e" dy C du
y J log it
The latter integral f [I/log u~\du is a transcendental function which
is called the integral logarithm.
119. Miscellaneous integrals. Let us consider an integral of the form
inx, cos a;) eta,
where / is an integral function of sin x and cos x. Any term of
this integral is of the form
where m and n are positive integers. We have seen above that the
product sinmx cos"x may be replaced by a sum of sines and cosines
of multiples of x. Hence it only remains to study the following
two types :
I eaxcosbxdx, I eaxsinbxdx.
Integrating each of these by parts, we find the formulae
C » eaxsinbx a T
I e"* cos bx dx = ; I e"* sin bx dx ,
J b b J
/pQZC QQg Jjnf* ft, I
eax sin bx dx = — h T I e"x cos &x dx.
b b J
Hence the values of the integrals under consideration are
e"x (a cos bx -f b sin bx)
eax cos bx dx = — * r: L >
a* + 6*
eax(a sin bx — b cos bx)
e"* sm bx dx = — » 5 —„ -•
a2 + i2
Among the integrals which may be reduced to the preceding
types we may mention the following cases :
I /(log x) x™ dx , I /(arc sin x) dx ,
I /(x) arc sin x dx , I /(x) arc tan x dx ,
246 INDEFINITE INTEGRALS [V, EM.
where / denotes any integral function. In the first two cases we
should take log x or arc sin x as the new variable. In the last
two we should integrate by parts, taking /(#) dx as the differential
of another polynomial F(x), which would lead to types of integrals
already considered.
EXERCISES
1. Evaluate the indefinite integrals of each of the following functions :
z* - x8 - 3x2 - x I + Vl + x
(x* + I)2 x (x3 + I)3 (x2 + I)8 i _ Vx
3 .
1 1 + VI + x 1 x
1 + x + Vl + x2 i_vT^~x' Vx + Vx + 1 + Vx(x + 1) ' cos2x'
x2 P
XC*COSX, , TV tan T
/ «t* * L til 1 X •
V a -f x" + 2
2. Find the area of the loop of the folium of Descartes :
x3 + y3 — Saxy — 0.
3. Evaluate the integral fy dx, where x and y satisfy one of the following
identities :
(x2 - a2)2 - ay2 (2y + 3a) = 0 , y2 (a - x) = x3 , y (x2 + y2) = a (y2 - x2) .
4. Derive the formulae
xcosnx
+ (7.
[EDLER.]
/. , , , sin" x cos nx
snv'-^x cos(n + l)xdx = — f- C,
n
/sinnxsinnx
smn-lx sin (n + l)xdx = \-C,
n
/cos"x sinnx
cos»-1xcos(n + l)xdx= — + C,
/cosnx cos
cos"-1 x sin (n + l)xdx =
5. Evaluate each of the following pseudo-elliptic integrals :
/(l + x2)dx C (l-x2)dx
,.•> .
6. Reduce the following integrals to elliptic integrals :
E(x)dx
Va(l + x6) + 6x(l + x*) + cx2(l + x2) + dx*
R(x)dx
Va(l + x8) + 6x2 (1 + x*) + ex*'
where R(x) denotes a rational function.
V, E».] EXERCISES 247
7*. Let a, 6, c, d be the roots of an equation of the fourth degree P(x) = 0.
Then there exist three involutory relations of the form
Mix" + Ni
t <= 1, 2, 3,
which interchange the roots by pairs. If the rational function f(x) satisfies the
identity
the integral f[f(x)/ VP (x)] dx is pseudo-elliptic (see Bulletin de la SocMM matM-
matique, Vol. XV, p. 106).
8. The rectification of a curve of the type y = Ax* leads to an integral of
a binomial differential. Discuss the cases of integrability.
9. If a > 1, show that
+ 1 ,
dx
/_, (a - x) Vl - x2 Va2 - 1
Hence deduce the formula
1
X
t
ndx _ 1. 3. 5-- -(2re - 1)
l - x2 ~ 2.4.6..-2n
10. If ^1C - J52 > 0, show that
X*"* dx _ 1 . 3 . 5 • • • (2ft - 3)
x (^lx2 + 2Bz + C)» ~ 2 . 4 . 6 • • • (2n - 2)
[Apply the reduction formula of § 104.]
11. Evaluate the definite integral
sin2xdx
C
Jo
1 + 2a cos x + a2 '
12. Derive the following formulae :
dx 1
C
JL
LI Vl - 2ax + a2 Vl - 2/3x
+ 1
- ax)(l - /3x) dz TT 2 -
- 2ax + a2)(l - 2^x + p?) Vl - x2 2 1 -
13*. Derive the formula
xm~ldx it
f
I/O
, *1 nsinM*
n
where m and n are positive integers (m<n). [Break up the integrand into
partial fractions.]
248 INDEFINITE INTEGRALS [V, Exs
14. From the preceding exercise deduce the formula
/
i/O
xn-}dx it
1 + x
15. Setting Ip<q = ftq(t + l)pdt, deduce the following reduction formulae :
(p + q + l)Ip,q = *» + !(« + !)*+!»/„_,,„
(P ~ !)*-*,« = i9 + l(t + I)1-* ~ (2 + q-p)I-P + i,g,
and two analogous formulae for reducing the exponent q.
16. Derive formulae of reduction for the integrals
7 _ C x»dx z _ C dx
J V2x2 + 2Bx + C ~J (x - a)* V^4z2 + 2Bx + C '
17*. Derive a reduction formula for the integral
C xndx
J0 vnr^'
Hence deduce a formula analogous to that of Wallis for the definite integral
18. Has the definite integral
£
1 dx
o vl-x*
/
Jo
dx
^ ^ 1+ x4 sin2x
a finite value ?
19. Show that the area of a sector of an ellipse bounded by the focal axis
and a radius vector through the focus is
P2 r
= ^j
tSQ
(1 + e cos w)'2
where p denotes the parameter W/a and e the eccentricity. Applying the gen
eral method, make the substitutions tan w/2 = t, t = u V(l + e)/(l — e) succes
sively, and show that the area in question is
A = ab I arc tan u — e )
V 1 + M«/
Also show that this expression may be written in the form
ab .
A = — (* - e sin
where <f> is the eccentric anomaly. See p. 406.
20. Find the curves for which the distance NT, or the area of the triangle
MNT, is constant (Fig. 3, p. 31). Construct the two branches of the curve.
[Licence, Paris, 1880; Toulouse, 1882.]
V, Exs.j EXERCISES 249
21*. Setting
X2n + l /.I
An = — - / (1 — z2)"cosxzdzt
2 . 4 . 6 . • • 2n Jo ^
derive the recurrent formula
From this deduce the formulae
AZP = Utp sin x + Vzp cos z ,
•A.ip + 1 = Uzp + 1 sin x + Vzp + 1 cos x ,
where UZP, V%p, UVP + I, V^^ + i are polynomials with integral coefficients, and
where £72p and U^p + \ contain no odd powers of x. It is readily shown that
these formulae hold when n = 1, and the general case follows from the above
recurrent formula.
The formula for Aop enables us to show that n2 is incommensurable. For if
we assume that 7T2/4 = b/a, and then replace x by ir/2 in A^p , we obtain a
relation of the form
a 2 . 4 . 6 • • • 4p
f V -*)»»• COB ^<b,
Jo 2
where HI is an integer. Such an equation, however, is impossible, for the right*
nand side approaches zero as p increases indefinitely.
CHAPTER VI
DOUBLE INTEGRALS
I. DOUBLE INTEGRALS METHODS OF EVALUATION
GREEX'S THEOREM
120. Continuous functions of two variables. Let 2 = f(x, y} be a
function of the two independent variables x and y which is contin
uous inside a region A of the plane which is bounded by a closed
contour C, and also upon the contour itself. A number of proposi
tions analogous to those proved in § 70 for a continuous function
of a single variable can be shown to hold for this function. For
instance, given any positive number c, the region A can be divided into
subregions in such a, way that the difference between the values of z at
any tivo points (x, y), (x', y') in the same subregion is less than e.
We shall always proceed by means of successive subdivisions as
follows : Suppose the region A divided into subregions by drawing
parallels to the two axes at equal dis
tances 8 from each other. The corre
sponding subdivisions of A are either
squares of side 8 lying entirely inside C,
or else portions of squares bounded in
part by an arc of C. Then, if the prop
osition were untrue for the whole region
A, it would also be untrue for at least
x one of the subdivisions, say A^. Sub
dividing the subregion Al in the same
manner and continuing the process indefinitely, we would obtain a
sequence of squares or portions of squares A, Alf • ••, An, •••, for
which the proposition would be untrue. The region An lies between
the two lines x = an and x = bn, which are parallel to the y axis,
and the two lines y = cn, y = dn, which are parallel to the x axis.
As n increases indefinitely an and bn approach a common limit A,
and cn and dn approach a common limit /A, for the numbers «„,
for example, never decrease and always remain less than a fixed
number. It follows that all the points of AH approach a limiting
250
FIG. 23
VI, §120] INTRODUCTION GREEN'S THEOREM 251
point (\, //,) which lies within or upon the contour C. The rest of
the reasoning is similar to that in § 70 ; if the theorem stated were
untrue, the function f(x, y) could be shown to be discontinuous at
the point (A, /*), which is contrary to hypothesis.
Corollary. Suppose that the parallel lines have been chosen
so near together that the difference of any two values of z in any
one subregion is less than e/2, and let -^ be the distance between
the successive parallels. Let (x, y*) and (x1, y') be two points inside
or upon the contour C, the distance between which is less than rj.
These two points will lie either in the same subregion or else in
two different subregions which have one vertex in common. In
either case the absolute value of the difference
f(x,y}-f(x',y<}
cannot exceed 2e/2 = c. Hence, given any positive number e, another
positive number 17 can be found such that
\f(x, y}-f(x>, y')\<€
whenever the distance between the two points (x, y*) and (x', y'), which
lie in A or on the contour C, is less than rj. In other words, any func
tion which is continuous in A and on its boundary C is uniformly
continuous.
From the preceding theorem it can be shown, as in § 70, that every
function which is continuous in A (inclusive of its boundary) is neces
sarily finite in A. If M be the upper limit and m the lower limit of
the function in A, the difference M — m is called the oscillation. The
method of successive subdivisions also enables us to show that the
function actually attains each of the values m and M at least once
inside or upon the contour C. Let a be a point for which z = m
and b a point for which z = M, and let us join a and b by a broken
line which lies entirely inside C. As the point (x, y) describes this
line, z is a continuous function of the distance of the point (x, y)
from the point a. Hence z assumes every value p. between m and
M at least once upon this line (§ 70). Since a and b can be joined
by an infinite number of different broken lines, it follows that the
f unction f(x, ?/) assumes every value between m and M at an infinite
number of points which lie inside of C.
A finite region A of the plane is said to be less than I in all its
dimensions if a circle of radius I can be found which entirely
encloses A. A variable region of the plane is said to be infinitesimal
252 DOUBLE INTEGRALS [VI, § 121
in all its dimensions if a circle whose radius is arbitrarily preas-
signed can be found which eventually contains the region entirely
within it. For example, a square whose side approaches zero or au
ellipse both of whose axes approach zero is infinitesimal in all its
dimensions. On the other hand, a rectangle of which only one side
approaches zero or an ellipse only one of whose axes approaches zero
is not infinitesimal in all its dimensions.
121. Double integrals. Let the region A of the plane be divided
into subregions ax, a2, • • -, an in any manner, and let u>,- be the area of
the subregion a,-, and M{ and m,- the limits of /(a;, y) in at. Consider
the two sums
each of which has a definite value for any particular subdivision
of A. None of the sums £ are less than m£l* where ft is the area of
the region A of the plane, and where m is the lower limit of f(x, y)
in the region A ; hence these sums have a lower limit /. Likewise,
none of the sums s are greater than 3/ft, where M is the upper limit
of f(x, y) in the region A ; hence these sums have an upper limit /'.
Moreover it can be shown, as in § 71, that any of the sums S is
greater than or equal to any one of the sums s; hence it follows
that
/>/'.
If the function f(x, y) is continuous, the sums S and s approach
a common limit as each of the subregions approaches zero in all its
dimensions. For, suppose that rj is a positive number such that the
oscillation of the function is less than c in any portion of A which
is less in all its dimensions than 77. If each of the subregions aif
a2, • • •, an be less in all its dimensions than rj, each of the differences
Mi — nii will be less than e, and hence the difference S — s will be
less than eft, where ft denotes the total area of A. But we have
S -s = S-I+I- /' + /'-*,
where none of the quantities S — 7, 7 —/',/' — s can be negative.
Hence, in particular, / — 7'<eft; and since e is an arbitrary posi
tive number, it follows that 7 = /'. Moreover each of the numbers
S — / and / — s can be made less than any preassigned number by
*If f(x, y) is a constant k, M = m = Mf = m< • = k, and S = s = mft = MQ. —
TRANS.
VI, §121] INTRODUCTION GREEN'S THEOREM 253
a proper choice of e. Hence the sums >S and s have a common limit
/, which is called the double integral of the function f(x, y\ extended
over the region A. It is denoted by the symbol
-//'
J J(A)
and the region A is called the field of integration.
If (|,., 77,) be any point inside or on the boundary of the sub-
region «„ it is evident that the sum 2/(£, 77,-) to, lies between the two
sums S and s or is equal to one of them. It therefore also
approaches the double integral as its limit whatever be the method
of choice of the point (£,-, 77,).
The first theorem of the mean may be extended without difficulty (
to double integrals. Let f(x, y) be a function which is continuous
in A, and let <f>(x, y) be another function which is continuous and
which has the same sign throughout A. For definiteness we shall
suppose that <$>(x, y) is positive in A. If M and m are the limits of
f(x, y] in A, it is evident that*
Adding all these inequalities and passing to the limit, we find the
formula
I I Ax> y)<Kx> y)dxdy = nl I *(*»
J J(A) J J(A)
where /x lies between M and m. Since the function f(x, y) assumes
the value /A at a point (£, 77) inside of the contour C, we may write
this in the form
(1) ff f(x, y)4>(x, y)dxdy =/(£, 77) \\ 4>(x,
J J(A) J J<A
which constitutes the law of the mean for double integrals. If
<£(a:, y~) = 1, for example, the integral on the right, ffdx dy, extended
over the region A, is evidently equal to the area O of that region.
In this case the formula (1) becomes
(2) f f f(x, y) dx dy = fl/(£ 77) .
J J(A)
* If f(x, y) is a constant k, we shall have M = m = k, and these inequalities become
equations. The following formula holds, however, with M= k. — TRANS. .
254 DOUBLE INTEGRALS [VI, §122
122. Volume. To the analytic notion of a double integral corre
sponds the important geometric notion of volume. Let f(x, y) be
a function which is continuous inside and upon a closed contour C.
We shall further suppose for definiteness that this function is posi
tive. Let S be the portion of the surface represented by the equa
tion z =f(x, y) which is bounded by a curve T whose projection
upon the xy plane is the contour C. We shall denote by E the por
tion of space bounded by the xy plane, the surface S, and the cylinder
whose right section is C. The region A of the xy plane which is
bounded by the contour C being subdivided in any manner, let a,- be
one of the subregions bounded by a contour ct-, and o>f the area of
this subregion. The cylinder whose right section is the curve c{ cuts
out of the surface 5 a portion st- bounded by a curve y{. Let p{ and
Pf be the points of s{ whose distances from the xy plane are a mini
mum and a maximum, respectively. If planes be drawn through
these two points parallel to the xy plane, two right cylinders are
obtained which have the same base o>0 and whose altitudes are the
limits M{ and mt- of the function /(cc, y) inside the contour c,, respec
tively. The volumes Vt and v{ of these cylinders are, respectively,
co, Mf and w,-m,-.* The sums S and s considered above therefore repre
sent, respectively, the sums 2Ff and ^vt of these two types of cylin
ders. We shall call the common limit of these two sums the volume
of the portion E of space. It may be noted, as was done in the case
of area (§ 78), that this definition agrees with the ordinary concep
tion of what is meant by volume.
If the surface S lies partly beneath the xy plane, the double integral
•will still represent a volume if we agree to attach the sign — to the
volumes of portions of space below the xy plane. It appears then that
every double integral represents an algebraic sum of volumes, just as
a simple integral represents an algebraic sum of areas. The limits of
integration in the case of a simple integral are replaced in the case of a
double integral by the contour which encloses the field of integration.
123. Evaluation of double integrals. The evaluation of a double
integral can be reduced to the successive evaluations of two simple
integrals. Let us first consider the case where the field of integration
*By the volume of a right cylinder we shall understand the limit approached by
the volume of a right prism of the same height, whose base is a polygon inscribed in
a right section of the cylinder, as each of the sides of this polygon approaches zero.
[This definition is not necessary for the argument, but is useful in showing that the
definition of volume in general agrees with our ordinary conceptions. —TRANS.]
VI, §123] INTRODUCTION GREEN'S THEOREM
255
is a rectangle R bounded by the straight lines x = x0, x = X,
y = y0, y = Y, where x0 < X and y0 < F. Suppose this rectangle
to be subdivided by parallels to the two axes x = x(, y = yk
(i = 1, 2, • • •, n ; k = 1, 2, • • •, TO). The area of the small rectangle
Rik bounded by the lines x = #,•_„ a; = or,-, y = yk_l, y = yk is
Hence the double integral is the limit of the sum
where (£,*, r)ik) is any point
inside or upon one of the
sides of Rik.
We shall employ the inde-
termination of the points
(£it> Vik) in order to simplify
the calculation. Let us re
mark first of all that if /(a-)
is a continuous function in
the interval (a, £), and if the interval (a, b~) be subdivided in any
manner, a value £ can be found in each subinterval (xi_l, #,-) such
that
y
0
Y
^
vk
t/2
2/i
2/o
B
0
a
'o •
c,;
58 2
i-1 '
Bf «
;»-!
X *
For we need merely apply the law of the mean for integrals to each of
the subintervals (a, Xj), (a^, a-2), • • • , (#„_,, &) to find these values of &.
Now the portion of the sum S which arises from the row of rec
tangles between the lines x = a:,-.! and x = x( is
^2)G/2 - //i) + ' ' '
Let us take ^a = |/2 = • • • = ^im = x^l} and then choose i;n, 77,2, • • •
in such a way that the sum
/(*,•-!» i?n)(yi - 2/0) +/(»,•-!> i7,-s)(y» - yO H —
is equal to the integral f!/Yf(xi_-[, y)dy, where the integral is to be
evaluated under the assumption that xi_l is a constant. If we pro
ceed in the same way for each of the rows of rectangles bounded by
two consecutive parallels to the y axis, we finally find the equation
(5) S = *(or0)(x1 -
256 DOUBLE INTEGRALS [VI, § 123
where we have set for brevity
*0«0 = / f(x> y)dy-
JV*
This function 4>(x), defined by a definite integral, where x is con
sidered as a parameter, is a continuous function of x. As all the
intervals xi — xi_l approach zero, the formula (5) shows that S
approaches the definite integral
..r
(#) dx .
Jx.
Hence the double integral in question is given by the formula
(6) f f /(*> !/) dxdy = f dx ff(x, y) dy .
J J(,R) J*t «A0
In other words, in order to evaluate the double integral, the function
f(x, y) should first be integrated between the limits y0 and Y, regard
ing x as a constant and y as a variable ; and then the resulting func
tion^ which is a function of x alone, should be integrated again between
the limits x0 and X.
If we proceed in the reverse order, i.e. first evaluate the portion
of S which comes from a row of rectangles which lie between two
consecutive parallels to the x axis, we find the analogous formula
/ I /(*> y)dxdy = I dy I f(x, y)dx.
J J(R) Jy0 Jx9
A comparison of these two formulae gives the new formula
x»;r />r /->Y ^x
I dx I f(x, y) dy = I dy I f(x, y)dx}
Jx0 JV(S Jy0 Jx0
which is called the formula for integration under the integral sign.
An essential presupposition in the proof is that the limits x0, X, y0, Y
are constants, and that the function f(x, y} is continuous throughout
the field of integration.
Example. Let z = xy/a. Then the general formula gives
cc ^-
/JU>*
VI, §123] INTRODUCTION GREEN'S THEOREM 257
In general, if the function f(x, y*) is the product of a function of x
alone by a function of y alone, we shall have
/ / <t>(x)$(y)dxdy = I $(x)dx x I
J J(R) Jxn J'.I0
The two integrals on the right are absolutely independent of each
other.
Franklin * has deduced from this remark a very simple demonstration of cer
tain interesting theorems of Tchebycheff. Let <f>(x) and f (x) be two functions
which are continuous in an interval (a, b), where a < b. Then the double integral
extended over the square bounded by the lines x = o, x = 6, y = a, y = b is equal
to the difference
2(6 - a) C <t>(x)\l/(x)dx — 2 C (p(x)dx x C \f/(x)dx.
Ja Ja */a
But all the elements of the above double integral have the same sign if the two
lunctions 0(z) and ^(z) always increase or decrease simultaneously, or if one of
them always increases when the other decreases. In the first case the two func
tions (f>(x) — (f>(y) and \f/(x) — ^(y) always have the same sign, whereas they have
opposite signs in the second case. Hence we shall have
(b -a) C <j>(x)t(x)dx > C $(x)dx x C ^(x)dx
Ja J a Ja
whenever the two functions <j>(x) and \f/(x) both increase or both decrease through
out the interval (a, b). On the other hand, we shall have
(b- a) f <f>(x)^(x)dx < f 0(z)dz x f
Ja Ja Ja
whenever one of the functions increases and the other decreases throughout the
interval.
The sign of the double integral is also definitely determined in case 0(z) = ^(z),
for then the integrand becomes a perfect square. In this case we shall have
(b -a]
whatever be the function 0(z), where the sign of equality can hold only when
<p(x) is a constant.
The solution of an interesting problem of the calculus of variations may be
deduced from this result. Let P and Q be two fixed points in a plane whose
coordinates are (a, A) and (6, B), respectively. Let y =/(z) be the equation of
any curve joining these two points, where /(z), together with its first derivative
* American Journal of Mathematics, Vol. VII, p. 77.
258
DOUBLE INTEGRALS
[VI, §124
/'(x), is supposed to be continuous in the interval (a, b). The problem is to
find that one of the curves y=f(x) for which the integral f^ y'2 dx is a
minimum. But by the formula just found, replacing <f>(x) by y" and noting
that /(a) = A and f(b) = B by hypothesis, we have
(b-a) Cy'*dx^(B-
•) a
A
B'
The minimum value of the integral is therefore (B — A)*/(b — a), and that value
is actually assumed when y' is a constant, i.e. when the curve joining the two
fixed points reduces to the straight line PQ.
124. Let us now pass to the case where the field of integration is
bounded by a contour of any form whatever. We shall first suppose
that this contour is met in at most two points by any parallel to the
y axis. We may then suppose that it is composed of two straight
lines x = a and x = b (a < 6)
and two arcs of curves APB
and A'QB' whose equations are
YI = <£: (cc) and F2 = $2 (#)> re~
spectively, where the functions
<f>! and <£2 are continuous be
tween a and b. It may happen
that the points A and A' coin
cide, or that B and B' coin
cide, or both. This occurs, for
instance, if the contour is a convex curve like an ellipse. Let us
again subdivide the field of integration R by means of parallels to
the axes. Then we shall have two classes of subregions : regular if
they are rectangles which lie wholly within the contour, irregular
if they are portions of rectangles bounded in part by arcs of the
contour. Then it remains to find the limit of the sum
FIG. 25
where o> is the area of any one of the subregions and (£, rj) is a point
in that subregion.
Let us first evaluate the portion of S which arises from the row
of subregions between the consecutive parallels x = xi_l, x = x(.
These subregions will consist of several regular ones, beginning
with a vertex whose ordinate is y' ^ Yt and going to a vertex whose
ordinate is y" ^ Y2, and several irregular ones. Choosing a suitable
point (£, 77) in each rectangle, it is clear, as above, that the portion
of S which comes from these regular rectangles may be written in
the form
VI, §124] INTRODUCTION GREEN'S THEOREM 259
(«,-_ i, y)dy.
Suppose that the oscillation of each of the functions <£i(#) and <£2 (a;)
in each of the intervals (x{_l} xt) is less than 8, and that each of the
differences yk — yk_l is also less than 8. Then it is easily seen that
the total area of the irregular subregions between x = xi_1 and x = xt
is less than 48(xt- — £;_,), and that the portion of S which arises
from these regions is less than 4:HB(x{ — xt_^) in absolute value,
where H is the upper limit of the absolute value of f(x, y) in the
whole field of integration. On the other hand, we have
XV" f*Yt s*Yi ny»
/(*<-i> y)dy = I /(»,-i, y}dy+ I + / ,
Jf\ Jy J Y*
and since \Yl — y'\ and |F2 — y"\ are each less than 28, we may write
f/fa-u y)dy= C f(xM)
Jy' J }',
The portion of S which arises from the row of subregions under
consideration may therefore be written in the form
where 0,- lies between — 1 and + 1. The sum SH8'2,Oi(xi — ar£_j) is
less than 87/8(6 — a) in absolute value, and approaches zero with 8,
which may be taken as small as we please. The double integral is
therefore the limit of the sum
where
Hence we have the formula
(7) f f f(*, V) dxdl/= fdx f f(x, y) dy.
J J(R) Ja J Yl
In the first integration x is to be regarded as a constant, but
the limits Yl and F2 are themselves functions of x and not
constants.
260 DOUBLE INTEGRALS [VI, § 124
Example. Let us try to evaluate the double integral of the function xy/a
over the interior of a quarter circle bounded by the axes and the circumference
X2 + yl _ #2 _ 0 .
The limits for x are 0 and R, and if x is constant, y may vary from 0 to VR2 — z2
Hence the integral is
p r^ ,» _.i(fe = /•«.(„. ^
Jo Jo <• Jo 2»L Jo J« 2»
The value of the latter integral is easily shown to be R*/8a.
When the field of integration is bounded by a contour of any form
whatever, it may be divided into several parts in such a way that
the boundary of each part is met in at most two points by a parallel
to the y axis. We might also divide it into parts in such a way that
the boundary of each part would be met in at most two points by
any line parallel to the x axis, and begin by integrating with respect
to x. Let us consider, for example, a convex closed curve which lies
inside the rectangle formed by the lines x = a, x = b, y = c, y = d,
upon which lie the four points A, B, C, D, respectively, for which x
or y is a minimum or a maximum.* Let yv = <#>i(«) and y2 = <£2(cc)
be the equations of the two arcs ACS and ADB, respectively, and
let o^ = 1/^1 (y) and x2 = \j/2 (y) be the equations of the two arcs CAD
and CBD, respectively. The functions <f>i(x~) and fa(x") are continu
ous between a and b, and i/^ (y) and i/^ (y) are continuous between c
and d. The double integral of a f unction /(x, y), which is continuous
inside this contour, may be evaluated in two ways. Equating the
values found, we obtain the formula
~& ~y2 ~fi -.T,
(8) I dx I f(x, y}dy = \ dy I /(or, y)dx.
•Jo, i/i/j t/c lyj-,
It is clear that the limits are entirely different in the two integrals.
Every convex closed contour leads to a formula of this sort. For
example, taking the triangle bounded by the lines y = 0, x = a,
y = x as the field of integration, we obtain the following formula,
which is due to Lejeune Dirichlet :
/ dx I f(x, y)dy=\dy\ f(x, y)dx.
Jo Jo Jo Jy
*The reader is advised to draw the figure.
VI, §125] INTRODUCTION GREEN'S THEOREM 261
125. Analogies to simple integrals. The integral JJf(t)dt, considered as a
function of x, has the derivative /(x). There exists an analogous theorem for
double integrals. Let f(x, y) be a function which is continuous inside a rec
tangle bounded by the straight lines x = a, x — A, y — b, y = J5,(a < A, b < B).
The double integral of /(x, y) extended over a rectangle bounded by the lines
x = a, x = X, y = b, y = F,(a < X < A, b < Y < .B), is a function of the coordi
nates X and Y of the variable corner, that is,
F(X,Y)= C'dxCf(x,y)dy.
J a J b
Setting *(x) = fb /(x, y) dy, a first differentiation with respect to X gives
— = *(X) = f /<-*, V)dy.
A second differentiation with respect to F leads to the formula
£2 F
(9)
The most general function u(X, Y) which satisfies the equation (9) is evi
dently obtained by adding to F(X, Y) a function z whose second derivative
d'2z/dXdY is zero. It is therefore of the form
(10) u(X, Y) = CAdx CYf(x, y) dy + <f(X) + f (F) ,
J a Jb
where <t>(X) and \1<(Y) are two arbitrary functions (see § 38). The two arbitrary
functions may be determined in such a way that u(X, Y) reduces to a given
function V(Y) when X = a, and to another given function U(X) when Y — b.
Setting X = a and then Y = b in the preceding equation, we obtain the two
conditions
V(Y) = 0(a) + *(F) , U(X) - t(X) + f (6) ,
whence we find
= F(F) - *(a) , *(&) = F(6) - 0(a) , <i>(X) = U(X) - F(6)
and the formula (10) takes the form
(11) u(X, Y) = (Xdx ( /(x, y) dy + U(X) + F(F) - F(6) .
*/ a J b
Conversely, if, by any means whatever, a function u(X, Y) has been found
which satisfies the equation (9), it is easy to show by methods similar to tne
above that the value of the double integral is given by the formula
(12) f dx f /(x, y)dy = u(X, Y) - u(X, b) - u(o, F) + u(a, b).
i/a «/6
This formula is analogous to the fundamental formula (6) on page 156.
The following formula is in a sense analogous to the formula for integration
by parts. Let A be a finite region of the plane bounded by one or more curves
262 DOUBLE INTEGRALS [VI, § 126
of any form. A function /(a;, y) which is continuous in A varies between its
minimum t0 and its maximum V. Imagine the contour lines /(x, y) = v drawn
where v lies between v0 and F, and suppose that we are able to find the area of
the portion of A for which /(x, y) lies between v0 and v. This area is a func
tion F(v) which increases with u, and the area between two neighboring contour
lines is F(v + A«) - F(v) = AvF'(v + 0Av). If this area be divided into infinitesi
mal portions by lines joining the two contour lines, a point (£, 77) may be found
in each of them such that /(£, i}) — v + 6A.v. Hence the sum of the elements
of the double integral / ffdxdy which arise from this region is
(V
It follows that the double integral is equal to the limit of the sum
that is to say, to the simple integral
v rv
v F'(v) dv = VF( F) - I F(v) dv .
This method is especially convenient when the field of integration is bounded
by two contour lines
/(x, y) = v0, /(x, y}= V.
For example, consider the double integral // Vl + x2 + y2 dx dy extended over
the interior of the circle x2 + j/2 = 1. If we set v = Vl + x2 + y'2, the field of
integration is bounded by the two contour lines v = 1 and v = \/2, and the
function F(v), which is the area of the circle of radius Vv'2 — 1, is equal to
7[(v* — I). Hence the given double integral has the value
/•v/iT 2ir
| 2itv2dv = — (2V5-1). *
J\ 3
The preceding formula is readily extended to the double integral
where F(o) now denotes the double integral ff<f>(x, y)dxdy extended over that
portion of the field of integration bounded by the contour line v =f(x, y).
126. Green's theorem. If the function f(x, y} is the partial deriva
tive of a known function with respect to either x or y, one of the
integrations may be performed at once, leaving only one indicated
integration. This very simple remark leads to a very important
formula which is known as Green's theorem.
* Numerous applications of this method are to be found in a memoir by Catalan
(Journal de Liouville, 1st series, Vol. IV, p. 233).
VI, §126] INTRODUCTION GREEN'S THEOREM 263
Let us consider first a double integral // cP/dy dx dy extended
over a region of the plane bounded by a contour C, which is met
in at most two points by any line parallel to the y axis (see Fig. 15,
p. 188).
Let A and B be the points of C at which x is a minimum and a
maximum, respectively. A parallel to the y axis between Aa and
Bb meets C in two points mx and mz whose ordinates are y-^ and yz,
respectively. Then the double integral after integration with respect
to y may be written
CCcP Cb C"^P C
JJ ~fydxd>J=J dxj ^ydy=j
But the two integrals fa P(x, y\)dx and fa P(x, y^)dx are line
integrals taken along the arcs AmlB and Am2B, respectively; hence
the preceding formula may be written in the form
(13)
where the line integral is to be taken along the contour C in the
direction indicated by the arrows, that is to say in the positive
sense, if the axes are chosen as in the figure. In order to extend
the formula to an area bounded by any contour we should proceed
as above (§ 94), dividing the given region into several parts for each
of which the preceding conditions are satisfied, and applying the for
mula to each of them. In a similar manner the following analogous
form is easily derived :
<u> // 1?
where the line integral is always taken in the same sense. Sub
tracting the equations (13) and (14), we find the formula
(15)
where the double integral is extended over the region bounded by C.
This is Green's formula ; its applications are very important. Just
now we shall merely point out that the substitution Q = x and
P = — y gives the formula obtained above (§ 94) for the area of a
closed curve as a line integral.
264 DOUBLE INTEGRALS [VI, § 127
II. CHANGE OF VARIABLES AREA OF A SURFACE
In the evaluation of double integrals we have supposed up to the
present that the field of integration was subdivided into infinitesimal
rectangles by parallels to the two coordinate axes. We are now going
to suppose the field of integration subdivided by any two systems of
curves whatever.
127. Preliminary formula. Let u and v be the coordinates of a point
with respect to a set of rectangular axes in a plane, x and y the coor
dinates of another point with respect to a similarly chosen set of
rectangular axes in that or in some other plane. The formulae
(16) x =f(u, v), y = <l>(u, v)
establish a certain correspondence between the points of the two
planes. We shall suppose 1) that the f unctions /(w, v) and </>(«, v),
together with their first partial derivatives, are continuous for all
points (u, v) of the uv plane which lie within or on the boundary of
a region Al bounded by a contour Cl; 2) that the equations (16)
transform the region AI of the uv plane into a region A of the
xy plane bounded by a contour C, and that a one-to-one correspond
ence exists between the two regions and between the two contours
in such a way that one and only one point of A 1 corresponds to any
point of A ; 3) that the functional determinant A = D(f, <£)/Z>(w, v)
does not change sign inside of Clt though it may vanish at certain
points of A i.
Two cases may arise. When the point (u, v) describes the con
tour Cl in the positive sense the point (x, y) describes the contour C
either in the positive or else in the negative sense without ever
reversing the sense of its motion. We shall say that the corre
spondence is direct or inverse, respectively, in the two cases.
The area fl of the region A is given by the line integral
Q = I
J(.C
taken along the contour C in the positive sense. In terms of the
new variables u and v defined by (16) this becomes
ft = ± I f(u, v) d<j>(u, v) ,
•Ac,)
where the new integral is to be taken along the contour Cl in the
positive sense, and where the sign -f- or the sign — should be taken
VI, § 127]
CHANGE OF VARIABLES
265
according as the correspondence is direct or inverse. Applying
Green's theorem to the new integral with x = u, v = y, P = fd<f>/du,
Q =/ d<f>/dv, we find
c/u cv
D(u, v)
A\
dudv ,
whence
or, applying the law of the mean to the double integral,
D(f, *)
(17)
n =
where (£, rf) is a point inside the contour Cl} and nl is the area of
the region Av in the uv plane. It is clear that the sign -f or the
sign — should be taken according as A itself is positive or negative.
Hence the correspondence is direct or inverse according as A is positive
or negative.
The formula (17) moreover establishes an analogy between func
tional determinants and ordinary derivatives. For, suppose that the
region A i approaches zero in all its dimensions, all its points approach
ing a limiting point (u, v~). Then the region A will do the same, and
the ratio of the two areas O and f^ approaches as its limit the abso
lute value of the determinant A. Just as the ordinary derivative is
the limit of the ratio of two linear infinitesimals, the functional
determinant is thus seen to be the limit of the ratio of two infinites
imal areas. From this point of view the formula (17) is the analogon
of the law of the mean for derivatives.
Remarks. The hypotheses which we have made concerning the correspondence
between A and AI are not all independent. Thus, in order that the correspond
ence should be one-to-one, it is necessary that A should not change sign in the
regional of the uv plane. For, suppose that A vanishes along a curve 71 which
divides the portion of AI where A is
positive from the portion where A is
negative. Let us consider a small arc
mini of -yi and a small portion of AI
which contains the arc mini. This
portion is composed of two regions a\
and a\ which are separated by mini
(Fig. 26).
When the point (u, v) describes the Fio. 26
region a\, where A is positive, the point
(x, y) describes a region a bounded by a contour mnpm, and the two contours
mi HI pi mi and mnpm are described simultaneously in the positive sense. When
the point («, v) describes the region af, where A is negative, the point (x, y)
266
DOUBLE INTEGRALS
[VI, § 128
describes a region a' whose contour nmqr is described in the negative sense as
n\m\q^n\ is described in the positive sense. The region a' must therefore
cover a part of the region a. Hence to any point (x, y) in the common part
nrm correspond two points in the uv plane which lie on either side of the
line mini.
As an example consider the transformation X = x, Y = y2, for which A = 2 y.
If the point (x, y) describes a closed region which encloses a segment a& of the
x axis, it is evident that the point (X, Y) describes two regions both of which
lie above the X axis and both of which are bounded by the same segment AB of
that axis. A sheet of paper folded together along a straight line drawn upon it
gives a clear idea of the nature of the region described by the point (X, Y}.
The condition that A should preserve the same sign throughout AI is not suf
ficient for one-to-one correspondence. In the example X = x2 — y2, Y = 2 xy,
the Jacobian A = 4 (x2 + y2) is always positive. But if (r, 6) and (.R, w) are the
polar coordinates of the points (x, y) and (X, F), respectively, the formulae of
transformation may be written in the form R = r2, u = 2 0. As r varies from a
to b (a < b) and 0 varies from OtO7T + a(0<a< Tf/2), the point (.R, u) describes
a circular ring bounded by two circles of radii a2 and b2. But to every value of
the angle u between 0 and 2a correspond two values of 6, one of which lies
between 0 and a, the other between it and it + a. The region described by the
point (X, Y) may be realized by forming a circular ring of paper which partially
overlaps itself.
128. Transformation of double integrals. First method. Retaining
the hypotheses made above concerning the regions A and Al and the
formulae (16), let us consider a function F(x, y) which is continuous
in the region A. To any subdivision of the region Al into subregions
alf a2, •••, an corresponds a subdivision of the region A into sub-
regions al} a2, • • •, an. Let to, and <r, be the areas of the two corre
sponding subregions a, and a,., respectively. Then, by formula (17),
(I),- = CTf
D(ui} vj
where «, and v{ are the coordinates of some point in the region a,.
To this point («,-, v,-) corresponds a point x, =/(«,-, v,), y,- = <£(w,-, ^.-)
of the region a,-. Hence, setting *(M, v) = -F[/(w, v), <£(w, v)], we
may write
D(f, <#
D(ui} Vi)
whence, passing to the limit, we obtain the formula
(18) f f F(x, y) dx dy = I I F[f(u, v}, <f>(u, v)
J J(A) J */Ui)
D(u, v)
dudv.
VI, § 128]
CHANGE OF VARIABLES
267
Hence to perform a transformation in a double integral x and y should
be replaced by their values as functions of the new variables u and v,
and dx dy should be replaced by | A | du dv. We have seen already
how the new field of integration is determined.
In order to find the limits between which the integrations should
be performed in the calculation of the new double integral, it is in
general unnecessary to construct the contour C\ of the new field
of integration Alf For, let us consider u and v as a system of
curvilinear coordinates, and let one of the variables u and v in the
formulae (16) be kept constant while the other varies. We obtain
in this way two systems of curves u = const, and v = const. By
the hypotheses made above, one and only one curve of each of these
families passes through any
given point of the region A.
Let us suppose for definite-
ness that a curve of the
family v = const, meets the
contour C in at most two
points MI and MI which cor
respond to values wt and uz
of u (HI < w2), and that each (<
of the (v) curves which meets 7 ^//^^f^T^
the contour C lies between
the two curves v — a and
v — b (a<b~). In this case
we should integrate first
with regard to u, keeping v constant and letting u vary from ^
to w2, where ul and uz are in general functions of v, and then inte
grate this result between the limits a and b.
The double integral is therefore equal to the expression
Fia. 27
f do f V[/(
U a \Ju.
«, V),
A change of variables amounts essentially to a subdivision of the
field of integration by means of the two systems of curves (u) and (v).
Let w be the area of the curvilinear quadrilateral bounded by the
curves (it), (u + du), (v), (v -f dv'), where du and dv are positive.
To this quadrilateral corresponds in the uv plane a rectangle whose
sides are du and dv. Then, by formula (17), w = | A(£, 77)) du dv, where
£ lies between u and u + du, and 77 between v and v + dv. The expres
sion | b.(u, v) | du dv is called the element of area in the system of
268 DOUBLE INTEGRALS [VI, § 129
coordinates (u, v~). The exact value of u> is o> = \\ A(M, v) \ + c \ du dv,
where c approaches zero with du and dv. This infinitesimal may be
neglected in finding the limit of the sum ^,F(x, y) w, for since A(M, v)
is continuous, we may suppose the two (u) curves and the two
(y) curves taken so close together that each of the e's is less in ab
solute value than any preassigned positive number. Hence the abso
lute value of the sum 2F(x, y^tdudv itself may be made less than
any preassigned positive number.
129. Examples. 1) Polar coordinates. Let us pass from rectangu
lar to polar coordinates by means of the transformation x = p cos w,
y = p sin GO. We obtain all the points of the xy plane as p varies
from zero to + oo and u> from zero to 2?r. Here A = p ; hence the
element of area is p da> dp, which is also evident geometrically. Let
us try first to evaluate a double integral extended over a portion of
the plane bounded by an arc AB which intersects a radius vector in
at most one point, and by the two straight lines OA and OB which
make angles ^ and to2 with the x axis (Fig. 17, p. 189). Let
R = <£(w) be the equation of the arc AB. In the field of integration
o> varies from ^ to o>2 and p from zero to R. Hence the double inte
gral of a function f(x, y) has the value
r» 2 CR
I «M /Oc
i/ojj i/O
cos GO, p sin CD) p dp .
If the arc AB is a closed curve enclosing the origin, we should
take the limits GOX = 0 and co2 = 2?r. Any field of integration can
be divided into portions of the preceding types. Suppose, for
instance, that the origin lies outside of the contour C of a given
convex closed curve. Let OA and OB be the two tangents from
the origin to this curve, and let RI =/1(<o) and 7?2 =/2(w) be the
equations of the two arcs ANB and A MB, respectively. For a
given value of o> between o^ and o>2, p varies from RI to 7?2, and
the value of the double integral is
•»• /•*»
6?(0 I /(p
, A
cos GO, p sin w) p dp.
2) Elliptic coordinates. Let us consider a family of confocal conies
CHANGE OF VARIABLES
269
where X denotes an arbitrary parameter. Through every point of the plane pass
two conies of this family, — an ellipse and an hyperbola, — for the equation (19)
FIG. 28
has one root X greater than c2, and another positive root p. less than c2, for any
values of x and y. From (19) and from the analogous equation where X is
replaced by p we find
(20)
y
V(X -
To avoid ambiguity, we shall consider only the first quadrant in the xy plane.
This region corresponds point for point in a one-to-one manner to the region of
the X/u plane which is bounded by the straight lines
X = c2, M = 0, n = cz.
It is evident from the formulae (20) that when the point (X, /u) describes the
boundary of this region in the direction indicated by the arrows, the point (a;, y)
describes the two axes Ox and Oy in the sense indicated by the arrows. The
transformation is therefore inverse, which is verified by calculating A :
= D(x, y) =
D(X, M)
130. Transformation of double integrals. Second method. We shall
now derive the general formula (18) by another method which
depends solely upon the rule for calculating a double integral. We
shall retain, however, the hypotheses made above concerning the
correspondence between the points of the two regions A and A±.
If the formula is correct for two particular transformations
x = f(n, v) ,
', v') ,
it is evident that it is also correct for the transformation obtained
by carrying out the two transformations in succession. This follows
at once from the fundamental property of functional determinants
(§30)
_ ,
D(u', v') D(u, v}' D(u', v')
270
DOUBLE INTEGRALS
[VI, § 130
Similarly, if the formula holds for several regions A, B, C, ••-, L,
to which correspond the regions Al9 B1) Clf • • •, L19 it also holds for
the region A -{- B -\- C + • • • + L. Finally, the formula holds if the
transformation is a change of axes :
x = x0 + x' cos a — y' sin a, y = y0 + x' sin a + y' cos a.
Here A = 1, and the equation
ff'
J J(A)
F(x, y} dx dy
F(x0 + x' cos « — y' sin a, y0 -f x' sin a + y' cos a) dx' dy1
is satisfied, since the two integrals represent the same volume.
We shall proceed to prove the formula for the particular trans
formation
(21^ x = d>(x' is1} 11 = ?/'
which carries the region A into a region A ' which is included between
the same parallels to the x axis, y = y0 and y = y±. We shall sup
pose that just one point of A corresponds to any given point of A' and
conversely. If a paral
lel to the x axis meets
the boundary C of the
region A in at most two
points, the same Avill be
true for the boundary
C" of the region A '. To
any pair of points m0
and m1 on C whose or-
x
dinates are each y cor
respond two points ?tt'0
and m{ of the contour C'. But the correspondence may be direct or
inverse. To distinguish the two cases, let us remark that if c<f>/dx' is
positive, x increases with x', and the points m0 and ml and m'0 and
m{ lie as shown in Fig. 29 ; hence the correspondence is direct. On
the other hand, if d<f>/dx' is negative, the correspondence is inverse.
Let us consider the first case, and let x0, Xi, x'n, x[ be the abscissae
of the points ra0, m^ m.'0, m{, respectively. Then, applying the for
mula for change of variable in a simple integral, we find
/in'.
FIG. 29
f
Jr
F(x,
(*', y'), y']
VI, §130] CHANGE OF VARIABLES 271
where y and y' are treated as constants. A single integration gives
the formula
rtfi r*1 ry< cx(
I dy\ F(x,y)dx=\ dy' I F^(
'J0 J^O J*» J'u
But the Jacobian A reduces in this case to d<f>/dx', and hence the
preceding formula may be written in the form
ff F(x, y}dxdy = ff F[A(x', y'}, y'] *\dx'dy'.
J J(A) J J(A')
This formula can be established in the same manner if d<f>/dx' is
negative, and evidently holds for a region of any form whatever.
In an exactly similar manner it can be shown that the trans
formation
(22) x = z', y = t(x', y')
leads to the formula
ff F(r, y)dxdU = ff F[.r', t(x', y')] I A | dx'dy',
J J(A) J J{4")
where the new field of integration .1' corresponds point for point tc
the region A.
Let us now consider the general formulae of transformation
(23) x =f(xl} yx), y =f, (x,, y,) ,
where for the sake of simplicity (a-, y) and (xl} y^) denote the coor
dinates of two corresponding points m and MI with respect to the
same system of axes. Let A and Al be the two corresponding regions
bounded by contours C and C1} respectively. Then a third point m',
whose coordinates are given in terms of those of m and Ml by the
relations x' = xl} y' = y, will describe an auxiliary region A', which
for the moment we shall assume corresponds point for point to each
of the two regions A and A^ The six quantities x, y, xlf yl} x', y'
satisfy the four equations
whence we obtain the relations
(24) x' = srl, y'=/i(*nyi),
which define a transformation of the type (22). From the equation
y' =/!(#', yj) we find a relation of the form yl = TT(X', y') ; hence
we may write
(25) x =f(x', yO = 4>(x', y'}, y = y'.
272 DOUBLE INTEGRALS [VI, § 131
The given transformation (23) amounts to a combination of the two
transformations (24) and (25), for each of which the general formula
holds. Therefore the same formula holds for the transformation (23).
Remark. We assumed above that the region described by the
point m' corresponds point for point to each of the regions A and
Av. At least, this can always be brought about. For, let us con
sider the curves of the region Al which correspond to the straight
lines parallel to the x axis in A. If these curves meet a parallel to
the y axis in just one point, it is evident that just one point m' of
A' will correspond to any given point m of A. Hence we need
merely divide the region At into parts so small that this condition
is satisfied in each of them. If these curves were parallels to the
y axis, AVO should begin by making a change of axes.
131. Area of a curved surface. Let S be a region of a curved sur
face free from singular points and bounded by a contour F. Let S
be subdivided in any way whatever, let s{ be one of the subregions
bounded by a contour yi} and let m^ be a point of st. Draw the tan
gent plane to the surface S at the point mi} and suppose s,- taken so
small that it is met in at most one point by any perpendicular to
this plane. The contour y,- projects into a curve y- upon this plane ;
we shall denote the area of the region of the tangent plane bounded
by yl by o-f . As the number of subdivisions is increased indefinitely
in such a way that each of them is infinitesimal in all its dimensions,
the sum 2o\- approaches a limit, and this limit is called the area of
the region S of the given surface.
Let the rectangular coordinates x, y, z of a point of S be given in
terms of two variable parameters u and v by means of the equations
(26) x =/(M, v), y = <f>(u, v), z = if/(u, v) ,
in such a way that the region S of the surface corresponds point for
point to a region R of the uv plane bounded by a closed contour C.
We shall assume that the functions /, <j>, and if/, together with their
first partial derivatives, are continuous in this region. Let R be
subdivided, let ri be one of the subdivisions bounded by a contour c,-,
and let wt- be the area of r{. To r{ corresponds on S a subdivision st
bounded by a contour yf. Let a-t be the corresponding area upon the
tangent plane defined as above, and let us try to find an expression
for the ratio o^/o^.
Let a{, (3^ yf be the direction cosines of the normal to the surface S
at a point mf(ict-, yt) z(~) of st which corresponds to a point (ui} Vf)
VI, § 131]
CHANGE OF VARIABLES
273
of rf. Let us take the point mt as a new origin, and as the new axes
the normal at m{ and two perpendicular lines m{X and mt Y in the
tangent plane whose direction cosines with respect to the old axes are
a', ft', y' and a", ft", y", respectively. Let X, Y, Z be the coordinates
of a point on the surface S with respect to the new axes. Then,
by the well-known formulae for transformation of coordinates, we
shall have
X = a' (x - *,.) + ?(y- yt) +?*(*- *,) ,
F = a"(x - x{) + ft"(y — yt) + y"(z — «f) ,
Z = or, (x — Xi) + fti (y — ?/,•) -f- y{ (z — «,-) .
The area tr,- is the area of that portion of the A'F plane which is
bounded by the closed curve which the point (X, F) describes, as
the point (u, v) describes the contour c£. Hence, by § 127,
p(x, y)
^-'Hswr^r
where u\ and v\ are the coordinates of some point inside of ci. An
easy calculation now leads us to the form
or, by the well-known relations between the nine direction cosines,
=s 3. < Oi
D(z,x) D(x,y)
D(u[, v't) T * D(u[, v§
' £><X> O
Applying the general formula (17), we therefore obtain the equation
*D(«4»O
where u\ and v\ are the coordinates of a point of the region r{ in the
uv plane. If this region is very small, the point (M,-, v£) is very near
the point (?*f, v,.), and we may write
T)fii >y\ T)fii v\ T)( f f\ Dff *r\
iJ\Jh Z) f^\J[t Z) . J'\z> •£) _ u\zi •c) i f
— \ ~r C«> ' ')
D(Ui, Vt)
u,
D(y,
where the absolute value of 6 does not exceed unity. Since the
derivatives of the functions /, <£, and ^ are continuous in the
274 DOUBLE INTEGRALS [VI, § i;;i
region R, we may assume that the regions rt have been taken so
small that each of the quantities e,-, e-, e-' is less than an arbitrarily
preassigned number rj. Then the supplementary term will certainly
be less in absolute value than 3^0, where O is the area of the
region R. Hence that term approaches zero as the regions st
(and Tf) all approach zero in the manner described above, and the
sum So-, approaches the double integral
(«)
(x, y)
D(u, v) D(u, v)
du dv ,
where a, ft, y are the direction cosines of the normal to the surface S
at the point (u, v~).
Let us calculate these direction cosines. The equation of the
tangent plane (§ 39) is
whence
a B y ±1
D(x, _ , (
D(u, v) D(u, v) D(u, v)
Choosing the positive sign in the last ratio, we obtain the formula
D(y>
, Q - '
D(u, v) P D(u, v)7 D(u, v)
.D(1
The well-known identity
(aft' - ba')2 + (be1 - c&')2 + (caf - ac')2
which was employed by Lagrange, enables us to write the quantity
under the radical in the form EG — F2, where
(27)
the symbol S indicating that a; is to be replaced by y and z succes
sively and the three resulting terms added. It follows that the area
of the surface S is given by the double integral
(28) A == / /
J J(B
- F*dudv.
(B)
VI, §132] CHANGE OF VARIABLES 275
The functions E, F, and G play an important part in the theory
of surfaces. Squaring the expressions for dx, dy, and dz and adding
the results, we find
(29) ds2 = dx2 + dy* + dz* = E du2 + 2Fdu dv + G dv2.
It is clear that these quantities E, F, and G do not depend upon
the choice of axes, but solely upon the surface S itself and the inde
pendent variables u and v. If the variables u and v and the sur
face 5 are all real, it is evident that EG — F2 must be positive.
132. Surface element. The expression V 'EG — F2 du dv is called the
element of area of the surface S in the system of coordinates (u, v).
The precise value of the area of a small portion of the siirface bounded
by the curves (u), (u + du), (v~), (v -\- dv) is (j\EG — F2 + t)dudv,
where e approaches zero with du and dv. It is evident, as above,
that the term e du dv is negligible.
Certain considerations of differential geometry confirm this result.
For, if the portion of the surface in question be thought of as a small
curvilinear parallelogram on the tangent plane to S at the point (u, v),
its area will be equal, approximately, to the product of the lengths
of its sides times the sine of the angle between the two curves (u)
and (v~). If we further replace the increment of arc by the differ
ential ds, the lengths of the sides, by formula (29), are ^/Edu and
^/Gdv, if du and dv are taken positive. The direction parameters of
the tangents to the two curves (u) and (v) are dx/du, dy/du, dz/du
and dx/dv, dy/dv, dz/dv, respectively. Hence the angle a between
them is given by the formula
^ dx dx
^ cu Gv F
COS a =
VI"
iV ( —
0 \du
whence sin a = V 'EG — .F2/V EG. Forming the product mentioned,
we find the same expression as that given above for the element of
area. The formula for cos a shows that F = 0 when and only when
the two families of curves (u) and (y~) are orthogonal to each other.
When the surface S reduces to a plane, the formulae just found
reduce to the formulae found in § 128. For, if we set if/(u, v) = 0,
we find
276
DOUBLE INTEGRALS
[VI, § 132
whence, by the rule for squaring a determinant,
dx ex
du dv
du dv
E F
F G
= EG - F2.
Hence ^EG — F2 reduces to |A|.
Examples. 1) To find the area of a region of a surface whose equa
tion is z = f(x} y} which projects on the xy plane into a region R in
which the function f(x, y), together with its derivatives p = df/dx and
q = df/dy, is continuous. Taking x and y as the independent vari
ables, we find E = \ + p2, F — pq, G = 1 + q2, and the area in ques
tion is given by the double integral
(30)
= ff
J J(R)
= ff
J J(R
(R) COS y
where y is the acute angle between the z axis and the normal to the
surface.
2) To calculate the area of the region of a surface of revolution
between two plane sections perpendicular to the axis of revolution.
Let the axis of revolution be taken as the z axis, and let z = f(x)
be the equation of the generating curve in the xz plane. Then the
coordinates of a point on the surface are given by the equations
z=f(p),
where the independent variables p and o> are the polar coordinates of
the projection of the point on the xy plane. In this case we have
p-), F=0, G = P*.
To find the area of the portion of the surface bounded by two plane
sections perpendicular to the axis of revolution whose radii are pt and
p2, respectively, p should be allowed to vary from Pl to p2 (pi< p2) and
«) from zero to 2?r. Hence the required area is given by the integral
and can therefore be evaluated by a single quadrature.
the arc of the generating curve, we have
ds* = rf2 + <fe« = <
If s denote
VI, §133] IMPROPER INTEGRALS 277
and the preceding formula may be written in the form
fp*
A = I 27Tp ds .
Jpl
The geometrical interpretation of this result is easy : 2jrp ds is
the lateral area of a frustum of a cone whose slant height is ds and
whose mean radius is p. Replacing the area between two sections
whose distance from each other is infinitesimal by the lateral area
of such a frustum of a cone, we should obtain precisely the above
formula for A.
For example, on the paraboloid of revolution generated by revolv
ing the parabola x9' = 2pz about the z axis the area of the section
between the vertex and the circular plane section whose radius is r is
III. GENERALIZATIONS OF DOUBLE INTEGRALS
IMPROPER INTEGRALS SURFACE INTEGRALS
133. Improper integrals. Let f(x, y) be a function which is con
tinuous in the whole region of the plane which lies outside a closed
contour F. The double integral of f(x, y) extended over the region
between F and another closed curve C outside of F has a finite value.
If this integral approaches one and the same limit no matter how
C varies, provided merely that the distance from the origin to the
nearest point of C becomes infinite, this limit is defined to be the
value of the double integral extended over the whole region
outside F.
Let us assume for the moment that the function f(x, y] has a
constant sign, say positive, outside F. In this case the limit of the
double integral is independent of the form of the curves C. For,
let Ci, Cz, •••, Cn, • • • be a sequence of closed curves each of which
encloses the preceding in such a way that the distance to the nearest
point of Cn becomes infinite with n. If the double integral /„ extended
over the region between F and Cn approaches a limit /, the same will
be true for any other sequence of curves C{, C^, •••, C'm, ••• which
satisfy the same conditions. For, if I'm be the value of the double
integral extended over the region between F and C'm, n may be
chosen so large that the curve Cn entirely encloses C'm, and wa
shall have /' < /„ < /. Moreover /' increases with m. Hence I'm
278 DOUBLE INTEGRALS [VI, § 133
has a limit /' < I. It follows in the same manner that I < I'. Hence
/' = I, i.e. the two limits are equal.
As an example let us consider a function f(x, y), which outside a
circle of radius r about the origin as center is of the form
where the value of the numerator \f/(x, y~) remains between two posi
tive numbers m and M. Choosing for the curves C the circles
concentric to the above, the value of the double integral extended
over the circular ring between the two circles of radii r and R is
given by the definite integral
\l/(p cos to, p sin o>)p dp
/, C
J,
It therefore lies between the values of the two expressions
By § 90, the simple integral involved approaches a limit as R
increases indefinitely, provided that 2a — 1 > 1 or a > 1. But it
becomes infinite with R if a < 1.
If no closed curve can be found outside which the function /(a;, y)
has a constant sign, it can be shown, as i» § 89, that the integral
ffffa y)dxdy approaches a limit if the integral // f(x, y) \dxdy
itself approaches a limit. But if the latter integral becomes infinite,
the former integral is indeterminate. The following example, due
to Cayley, is interesting. Let f(x, y) = sin (x2 + if), and let us inte
grate this function first over a square of side a formed by the axes
and the two lines x = a, y = a. The value of this integral is
ra ra
I dx I sin (a;2 + yz}dy
Jo Jo
= I sinx^dx x I cosy2dy+ I cosx*dx x I siny*dy.
Jo Jo Jo Jo
As a increases indefinitely, each of the integrals on the right has
a limit, by § 91. This limit can be shown to be V?r/2 in each case ;
hence the limit of the whole right-hand side is TT. On the other
hand, the double integral of the same function extended over the
quarter circle bounded by the axes and the circle x1 + y2 = R3 is
equal to the expression
/I, §134] IMPROPER INTEGRALS 279
7T
r* r
\ du \
*/0 «yo
which, as R becomes infinite, oscillates between zero and 7r/2 and
does not approach any limit whatever.
We should define in a similar manner the double integral of a
function f(x, y) which becomes infinite at a point or all along a line.
First, we should remove the point (or the line) from the field of
integration by surrounding it by a small contour (or by a contour
very close to the line) which we should let dimmish indefinitely.
For example, if the function f(x, y) can be written in the form
f(x „•) = _ ^(*> y)
»«
in the neighborhood of the point (a, b), where \}/(x, y) lies between
two positive numbers m and M, the double integral of f(x, y)
extended over a region about the point (a, b) which contains no
other point of discontinuity has a finite value if and only if a is
less than unity.
134. The function B(p, q). We have assumed above that the contour Cn
recedes indefinitely in every direction. But it is evident that we may also sup
pose that only a certain portion recedes to infinity. This is the case in the above
example of Cayley's and also in the following example. Let us take the function
/(x, y) = 4x2P-1y2«-1e-*2-!'2,
where p and q are each positive. This function is continuous and positive in the
first quadrant. Integrating first over the square of side a bounded by the axes
and the lines x — a and y = a, we find, for the value of the double integral,
C 2x2J>-ie-*2dx x Ca2y*i-ie-»*dy.
Jo Jo
Each of these integrals approaches a limit as a becomes infinite. For, by the
definition of the function T(p) in § 92,
T(p)= f +CV-ie-<(K,
Jo
whence, setting t = x2, we find
(31) r(p)= C + °°2x*P-*e-**dx.
Jo
Hence the double integral approaches the limit T(p) T(q) as a becomes infinite.
Let us now integrate over the quarter circle bounded by the axes and the
circle z2 + y2 = R2. The value of the double integral in polar coordinates is
f*R /*•
I 2p'2<-P + 'i)-le-P dp x I '
Jo J»
280 DOUBLE INTEGRALS [VI, § u»
As R becomes infinite this product approaches the limit
T(p + q)B(p, q),
where we have set
rr
(32) B(p,q) = C22cos?P-l<l>sin2<i-i<t>d<}>.
Jo
Expressing the fact that these two limits must be the same, we find the equation
(33) T(p)T(q) = r(p + q)B(p, q).
The integral B(p, q} is called Euler's integral of the first kind. Setting t = sin2 <£,
it may be written in the form
(34) B(p, q)= f t9-i(l - t)v-1 dt.
JQ
The formula (33) reduces the calculation of the function B(p, q) to the calcu
lation of the function T. For example, setting p — q = 1/2, we find
whence F(l/2) = vV. Hence the formula (31) gives
f.
2
In general, setting q = 1 — p and taking p between 0 and 1 , we find
tl-p) = f (~
Jo \ l
T(p)T(l-p) =
We shall see later that the value of this integral is jr/sin pit.
135. Surface integrals. The definition of surface integrals is analogous to that
of line integrals. Let S be a region of a surface bounded by one or more curves F.
We shall assume that the surface has two distinct sides in such a way that if one
side be painted red and the other blue, for instance, it will be impossible to pass
from the red side to the blue side along a continuous path which lies on the sur
face and which does not cross one of the bounding curves.* Let us think of S as
a material surface having a certain thickness, and let m and m' be two points
near each other on opposite sides of the surface. At m let us draw that half of
the normal mn to the surface which does not pierce the surface. The direction
thus defined upon the normal will be said, for brevity, to correspond to that side
of the surface on which m lies. The direction of the normal which corresponds
to the other side of the surface at the point m' will be opposite to the direction
just defined.
Let z = <f>(x, y) be the equation of the given surface, and let -S be a region of
this surface bounded by a contour F. We shall assume that the surface is met
in at most one point by any parallel to the z axis, and that the function 0(z, y)
* It is very easy to form a surface which does not satisfy this condition. We need
only deform a rectangular sheet of paper ABCD by pasting the side B C to the side AD
in such a way that the point C coincides with A and the point B with D.
VI, §135] SURFACE INTEGRALS 281
is continuous inside the region A of the xy plane which is bounded by the curve C
into which T projects. It is evident that this surface has two sides for which
the corresponding directions of the normal make, respectively, acute and obtuse
angles with the positive direction of the z axis. We shall call that side whose
corresponding normal makes an acute angle with the positive z axis the upper
side. Now let P(x, y, z) be a function ofi the three variables x, y, and z which
is continuous in a certain region of space which contains the region S of the sur
face. If z be replaced in this function by <£(x, y), there results a certain function
P [x, y, <p(x, y)] of x and y alone ; and it is natural by analogy with line integrals
to call the double integral of this function extended over the region A,
(35) f f P [x, y, 0(x, y)] dx dy ,
v <s (A)
the surface integral of the function P(x, y, z) taken over the region S of the given
surface. Suppose the coordinates x, y, and z of a point of S given in terms of two
auxiliary variables u and v in such a way that the portion S of the surface corre
sponds point for point in a one-to-one manner to a region R of the uv plane. Let
da- be the surface element of the surface S, and 7 the acute angle between the posi
tive z axis and the normal to the upper side of S. Then the preceding double
integral, by §§ 131-132, is equal to the double integral
(36) rr P(x, y, z)cos7d<r,
where x, y, and z are to be expressed in terms of u and v. This new expression
is, however, more general than the former, for cos 7 may take on either of two
values according to which side of the surface is chosen. When the acute angle 7
is chosen, as above, the double integral (35) or (36) is called the surface integral
(37)
extended over the upper side of the surface S. But if 7 be taken as the obtuse
angle, every element of the double integral will be changed in sign, and the new
double integral would be called the surface integral / / Pdxdy extended over the
lower side of <S. In general, the surface integral// Pdx dy is equal to ± the double
integral (35) according as it is extended over the upper or the lower side of S.
This definition enables us to complete the analogy between simple a^id double
integrals. Thus a simple integral changes sign when the limits are interchanged,
while nothing similar has been developed for double integrals. With the gen
eralized definition of double integrals, we may say that the integral///(x, y) dx dy
previously considered is the surface integral extended over the upper side of the
xy plane, while the same integral with its sign changed represents the surface
integral taken over the under side. The two senses of motion for a simple inte
gral thus correspond to the two sides of the xy plane for a double integral.
The expression (36) for a surface integral evidently does not require that the
surface should be met in at most one point by any parallel to the z axis. In the
same manner we might define the surface integrals
ff Q(z, y, z) dy dz, ffR(x' v* z) dz dx »
282
DOUBLE INTEGRALS
[VI, § 136
and the more general integral
f fp(x, y, z)dxdy + Q(x, y, z)dydz + R(x, y, z)dzdx.
This latter integral may also be written in the form
C f [Pcos? + Qcosa + JJcos/3]d<r,
where a, £, 7 are the direction angles of the direction of the normal which cor
responds to the side of the surface selected.
Surface integrals are especially important in Mathematical Physics.
136. Stokes* theorem. Let L be a skew curve along which the functions
P(x, y, z), Q(x, y, z), R(x, y, z) are continuous. Then the definition of the line
integral
Pdx + Qdy +Rdz
C
J(
(L)
taken along the line L is similar to that given in § 93 for a line integral taken
along a plane curve, and we shall not go into the matter in detail. If the curve L
is closed, the integral evidently may be broken up into the sum of three line inte
grals taken over closed plane curves. Applying Green's theorem to each of these,
it is evident that we may replace the line integral by the sum of three double
integrals. The introduction of surface integrals enables us to state this result in
very compact form.
Let us consider a two-sided piece S of a surface which we shall suppose for
definiteness to be bounded by a single curve P. To each side of the surface
corresponds a definite sense of direct motion along the contour r. We shall
assume the following convention : At any point M of the contour let us draw
that half of the normal Mn which corresponds to the side of the surface under
consideration, and let us imagine an observer with his head at n and his feet at M ;
we shall say that that is the positive sense
of motion which the observer must take in
order to have the region S at his left hand.
Thus to the two sides of the surface corre
spond two opposite senses of motion along
the contour F.
Let us first consider a region S of a sur
face which is met in at most one point by
any parallel to the z axis, and let us suppose
the trihedron Oxyz placed as in Fig. 30,
where the plane of the paper is the yz plane
and the x axis extends toward the observer.
To the boundary F of S will correspond a
closed contour C in the xy plane ; and these
two curves are described simultaneously in
the sense indicated by the arrows. Let
z = /(x, y) be the equation of the given surface, and let P(x, y, z) be a function
which is continuous in a region of space which contains S. Then the line inte
gral fr P(x, y, z) dx is identical with the line integral
VI, §136] SURFACE INTEGRALS 283
/ P[x, y, <p(x, y)]dx
«/(C)
taken along the plane curve C. Let us apply Green's theorem (§ 126) to this
latter integral. Setting
P(x, y) = P[z, y, <(>(x, y)]
for definiteness, we find
, y) _ dP dP_ d<fi _8P_ S
£y cy cz cy cy cz cosy
where or, /3, 7 are the direction angles of the normal to the upper side of S.
Hence, by Green's theorem,
-D, - ^j
P(x, y)dx
C C
= I
J J(A
dP
cz dV cos y
where the double integral is to be taken over the region A of the xy plane
bounded by the contour C. But the right-hand side is simply the surface
integral
— cos /3 cos 7 ) do-
dz dy I
extended over the upper side of S ; and hence we may write
f P(x, ?/, z)dx = / I --dzdx dxdy.
J(F) J J(S)cz dy
This formula evidently holds also when the surface integral is taken over the
other side of <S, if the line integral is taken in the other direction along F. And
it also holds, as does Green's theorem, no matter what the form of the surface
may be. By cyclic permutation of x, y, and z we obtain the following analogous
formulae :
f Q(x, y,z)dy=
J^
C R(x,y,z)dz= C i— dydz- — dzdx.
J<n J J(S)sy dx
Adding the three, we obtain Stokes' theorem in its general form :
/•
J P(x, y, z)dx + Q(z, y, z)dy + R(x, y, z)dz
dP\, ^(dE dQ\. /d ^
-- )dxdy + ( --- 2 }dydz + ( --- \dzdx.
ty/
The sense in which T is described and the side of the surface over which the
double integral is taken correspond according to the convention made above.
284 DOUBLE INTEGRALS [VI, § 137
IV. ANALYTICAL AND GEOMETRICAL APPLICATIONS
137. Volumes. Let us consider, as above, a region of space bounded
by the xy plane, a surface S above that plane, and a cylinder whose
generators are parallel to the z axis. We shall suppose that the
section of the cylinder by the plane z — 0 is a contour similar to
that drawn in Fig. 25, composed of two parallels to the y axis and two
curvilinear arcs APB and A 'QB'. If % — f(x, y) is the equation of the
surface S, the volume in question is given, by § 124, by the integral
rb rvt
V = \ dx I f(x, y)dy.
Jo. Jy,
Now the integral f "2/(o:, y}dy represents the area A of a section of
this volume by a plane parallel to the yz plane. Hence the preceding
formula may be written in the form
(39) V=f
U a
b
A dx.
The volume of a solid bounded in any way whatever is equal
to the algebraic sum of several volumes bounded as above. For
instance, to find the volume of a solid bounded by a convex closed
surface we should circumscribe the solid by a cylinder whose gen
erators are parallel to the z axis and then find the difference between
two volumes like the preceding. Hence the formula (39) holds for
any volume which lies between two parallel planes x = a and x = b
(a < I) and which is bounded by any surface whatever, where A
denotes the area of a section made by a plane parallel to the two
given planes. Let us suppose the interval (a, £) subdivided by the
points a, xl} xz, • • •, xn_l} b, and let A0, Au • • •, A;, • • • be the areas
of the sections made by the planes x = a, x = xlt • • -, respectively.
Then the definite integral $*k dx is the limit of the sum
The geometrical meaning of this result is apparent. For A,-_! (#,- — a;,-_i),
for instance, represents the volume of a right cylinder whose base is
the section of the given solid by the plane x = a-,-_i and whose height
is the distance between two consecutive sections. Hence the volume
of the given solid is the limit of the sum of such infinitesimal cylin
ders. This fact is in conformity with the ordinary crude notion of
volume.
VI, §138] APPLICATIONS 285
If the value of the area A be known as a fnnction of x, the vol
ume to be evaluated may be found by a single quadrature. As an
example let us try to find the volume of a portion of a solid of revo
lution between two planes perpendicular to the axis of revolution.
Let this axis be the x axis and let z = f(x) be the equation of the
generating curve in the xz plane. The section made by a plane par
allel to the yz plane is a circle of radius f(x). Hence the required
volume is given by the integral TT^ [/(x)]2e£r.
Again, let us try to find the volume of the portion of the ellipsoid
n n n
^4.^4. f! = i
a* b* c2
bounded by the two planes x = x0, x = X. The section made by a
plane parallel to the plane x = 0 is an ellipse whose semiaxes are
b Vl — x2/a2 and c Vl — x2/a2. Hence the volume sought is
a-A / Y8 -
*
V= I Trbc l-
cA0 \ «
To find the total volume we should set x = — a and X = a, which
gives the value %irabc.
138. Ruled surface. Prismoidal formula. When the area A is an integral
function of the second degree in z, the volume may be expressed very simply
in terms of the areas B and B' of the bounding sections, the area 6 of the mean
section, and the distance h between the two bounding sections. If the mean
section be the plane of yz, we have
V = C + °(te2 + 2mx + n) dx = 21 — + 2na.
J —a 3
But we also have
h = 2a, b = n, B = Itf + 2ma + n , B' = la* - 2ma + n ,
whence n = &, a = h/2, 2Za2 = B + B' — 26. These equations lead to the formula
(40) F=^[B + B' + 4&],
o
which is called the prismoidal formula.
This formula holds in particular for any solid bounded by a ruled surface and
two parallel planes, including as a special case the so-called prismoid.* For,
let y = ax + p and z = bx + q be the equations of a variable straight line, where
a, 6, p, and q are continuous functions of a variable parameter t which resume
their initial values when t increases from tQ to T. This straight line describes
* A prismoid is a solid bounded by any number of planes, two of which are paral
lel and contain all the vertices. —TRANS.
286
DOUBLE INTEGRALS
[VI, § 139
a ruled surface, and the area of the section made by a plane parallel to the plane
x = 0 is given, by § 94, by the integral
/•T
= I (ax + p)(b'x + q')dt,
Jto
where a', &', c', d' denote the derivatives of a, 6, c, d with respect to t. These
derivatives may even be discontinuous for a finite number of values between t0
and T, which will be the case when the lateral boundary consists of portions of
several ruled surfaces. The expression for A may be written in the form
rr rr rr
A = x2 / ab'dt + x I (aq' + pb')dt + \ pq'dt,
Jtn Jt0 Jt0
where the integrals on the right are evidently independent of x. Hence the
formula (40) holds for the volume of the given solid. It is worthy of notice that
the same formula also gives the volumes of most of the solids of elementary geometry '.
139. Viviani's problem. Let C be a circle described with a radius OA (= R)
of a given sphere as diameter, and let us try to find the volume of the portion
of the sphere inside a circular cylinder whose right section is the circle C.
Taking the origin at the center of the sphere, one fourth the required volume
is given by the double integral
-
- z» - 2/2 dxdy
extended over a semicircle described on OA as diameter. Passing to polar coor
dinates p and w, the angle u varies from 0 to it/2, and p from 0 to R cos w. Hence
we find
FIG. 31
v l rim m • « \* RSi* 2\
— = - I (R3 — R3 Bin* w) du = — ( --- ) •
4 3 Jo 3 \2 3/
If this volume and the volume inside the cylinder
which is symmetrical to this one with respect to
the z axis be subtracted from the volume of the
whole sphere, the remainder is
-
3 3 \2
Again, the area ft of the portion of the sur
face of the sphere inside the given cylinder is
dxdy.
Replacing p and q by their values — x/z and — y/z, respectively, and passing to
polar coordinates, we find
VI, § 140] APPLICATIONS 287
f ftRcoiui r> •, *
*• I Rf>d(> = 4 f 2 -
Jo V^2-,2 Jo
tt = 4fl2 f 2(l-sinw)dw = 4R2(- -
Jo a
Subtracting the area enclosed by the two cylinders from the whole area of the
sphere, the remainder is
'it
^
140. Evaluation of particular definite integrals. The theorems estab
lished above, in particular the theorem regarding differentiation
under the integral sign, sometimes enable us to evaluate certain defi
nite integrals without knowing the corresponding indefinite integrals
We proceed to give a few examples.
Setting
A = F(a\ = r*log(1 + aX^ dx
the formula for differentiation under the integral sign gives
dA _ log (1+a2) f xdx
da ~ 1 + a2
Breaking up this integrand into partial fractions, we find
x 1 Ix + a a
1+ax
whence
x dx log (1 + <*2) . «
• -fai
It follows that
a log(l4-«2)
- arc tan -
o/-i i
rfa 1+a2 2(1+ a2)
whence, observing that ,4 vanishes when a = 0, we may write
r log (i+<> ., rg a
A = \ 0/1 I 2\ ^ + I T~i - "2
Jo 2(1+ a2) J0 1+a2
Integrating the first of these integrals by parts, we finally find
A = - arc tan a log (1 + a2) .
m
288 DOUBLE INTEGRALS [VI, § 140
Again, consider the function xv. This function is continuous
when x lies between 0 and 1 and y between any two positive
numbers a and b. Hence, by the general formula of § 123,
/i /-»& /•»?> /* \
dx I xvdy = I dy I xydx.
t/a J a UQ
rl
Jo
But
>i
ipy flf —
I ,, I 1 I ,, i 1
'o Ly + IJ« y + L
hence the value of the right-hand side of the previous equation is
- = log
On the other hand, we have
r
£
log a;
whence
= log
log x
In general, suppose that P(x, y) and Q(x, ?/) are two functions
which satisfy the relation dP/dy = dQ/dx, and that x0, x^ y0, y^ are
given constants. Then, by the general formula for integration
under the integral sign, we shall have
pap p p0Q
c?o; I -—dy=\ dy \ -^-dx,
J* ty J,Jo JJ*0 ^
or
r*i r^
(41) 1 [P(x, y,}-P(x, y0)-]dx= I [Q(^15 y)- Q
•^^o ^"o
Cauchy deduced the values of a large number of definite inte
grals from this formula. It is also closely and simply related to
Green's theorem, of which it is essentially only a special case.
For it may be derived by applying Green's theorem to the line
integral fPdx + Qdy taken along the boundary of the rectangle
formed by the lines x = XQ, x — x^ y = y0, y = yl.
In the following example the definite integral is evaluated by a
special device. The integral
= I log (1 — 2a cos x + n-2) dx
Jo
VI, §110] APPLICATIONS 289
has a finite value if \a\ is different from unity. This function
F(<x) has the following properties.
1) F(- a) = F(a). For
F(- a) = I log (1 + 2a cos x + a-2) dx,
Jo
or, making the substitution x = TT — y,
F(- a) — I log (1 — 2a cos y + a2) dy = F(a) .
Jo
2) F(a2) = 2F(a). For we may set
2F(«) = F(«) + F(-a),
whence
2F(a) = I [log (1 — 2a cos x + a2) + log (1 + 2a COS x + a2)] dx
Jo
= I log (I - 2a* cos 2x + a*)dx.
Jo
If we now make the substitution 2x = y, this becomes
1 C"
2F(a) = - I \og(l-2a'icosy + a4)dy
* Jo
i r2*
+ - I log (1 - 2a2 cos y + a4) dy.
Making a second substitution y = lit — z in the last integral, we
find
X2rr f*n
\og(l-2a2cosy + a*)dy= I log (1 - 2«2cos z + a*)dz,
Jo
which leads to the formula
From this result we have, successively,
F(a) = \ F(a2) = \ F(a") = • • • = ±
If | a | is less than unity, a2" approaches zero as n becomes infinite.
The same is true of F(a2"), for the logarithm approaches zero.
Hence, if | a| < 1, we have F(a) = 0.
290 DOUBLE INTEGRALS [VI, § 141
If | a | is greater than unity, let us set a = I//?. Then we find
F(a) = f
Jo
2 cos x 1
= 1 log(l-2/?cos;r
Jo
where \ft\ is less than unity. Hence we have in this case
F(a) = — TT log ft'2 = TT log a2.
Finally, it can be shown by the aid of Ex. 6, p. 205, that F(± 1) = 0 ;
hence F(a) is continuous for all values of a.
141. Approximate value of logF(n + l). A great variety of devices may be
employed to find either the exact or at least an approximate value of a definite
integral. We proceed to give an example. We have, by definition,
n +00
T(n + l) = I xne-xdx.
Jo
The function xne~x assumes its maximum value nne~n f or x = n. As x increases
from zero to n, xne~x increases from zero to nne~n (n>0), and when x increases
from n to + oo, xne~x decreases from nnern to zero. Likewise, the function
nnerne-tt increases from zero to nnern as t increases from — oo to zero, and
decreases from nne~n to zero as t increases from zero to + oo. Hence, by the
substitution
n-* =
(42)
the values of x and t correspond in such a way that as t increases from — oo
to + oo, x increases from zero to + oo.
It remains to calculate dx/dt. Taking the logarithmic derivative of each side
of (42), we find
dx, 2tx
dt x — n
We have also, by (42), the equation
tfl = x — n — n log f - j •
W
For simplicity let us set x = n + z, and then develop log (1 + z/n) by Taylor's
theorem with a remainder after two terms. Substituting this expansion in the
value for <2, we find
nz2
[2
n
where 6 lies between zero and unity. From this we find, successively,
x -
VI, § 142] APPLICATIONS 291
whence, applying the formula for change of variable,
Y(n + 1) = 2nne-" •*/- C "e-^dt + 2n"ern C *V<2(1
\ / I/— 00 V— 00
The first integral is
» » • » ,_
e-<2dt = Vnr.
As for the second integral, though we cannot evaluate it exactly, since we do
not know 0, we can at least locate its value between certain fixed limits. For
all its elements are negative between — <x> and zero, and they are all positive
between zero and + oo. Moreover each of the integrals f_n, /0+cc is ^ess m
absolute value than /0+°° ter^dt = 1/2. It follows that
(43) F(n + 1) = \/2nnne~n( -
V*n/
where u lies between — 1 and + 1-
If n is very large, w/V2n is very small. Hence, if we take
= n"e-
as an approximate value of T(n + 1), our error is relatively small, though the
actual error may be considerable. Taking the logarithm of each side of (43), we
find the formula
(44) log r(n + 1) = (n + 1) logn - n + 1 log(27r) + e,
where e is very small when n is very large. Neglecting e, we have an expression
which is called the asymptotic value of logT(n + 1). This formula is inter
esting as giving us an idea of the order of magnitude of a factorial.
142. D'Alembert's theorem. The formula for integration under the integral
sign applies to any function /(x, y) which is continuous in the rectangle of inte
gration. Hence, if two different results are obtained by two different methods
of integrating the function /(x, y), we may conclude that the function /(x, y) is
discontinuous for at least one point in the field of integration. Gauss deduced
from this fact an elegant demonstration of d'Alembert's theorem.
Let F(z) be an integral polynomial of degree m in z. We shall assume for
definiteness that all its coefficients are real. Replacing z by p(cosw + isinw),
and separating the real and the imaginary parts, we have
F(z) = P + iQ,
where
P = A0pmcosm<,} + Aipm-lcos(m — !)&>
Q = A0pmsmmw + A\pm~l sin(m — 1) w
If we set V = arc tan (P/Q), we shall have
aP_paQ q^-P^
8V dp dp 8V 8u du
a/T = P*+ Q2 ~^ ~ P2 + Q2
and it is evident, without actually carrying out the calculation, that the second
derivative is of the form
&V M
292 DOUBLE INTEGRALS [VI, Exs.
where M is a continuous function of p and u. This second derivative can only
be discontinuous for values of p and <a for which P and Q vanish simultaneously,
that is to say, for the roots of the equation F(z) = 0. Hence, if we can show that
the two integrals
(4o)
r\ cRw* CR* c"*r*
I du I -—dp, I dp I — —du
Jo Jo dPdu Jo Jo dPCu
are unequal for a given value of R, we may conclude that the equation F(z) = 0
has at least one root whose absolute value is less than R. But the second inte
gral is always zero, for
f"
Jo
^
au = I —
and dV/dp is a periodic function of w, of period 2x. Calculating the first inte
gral in a similar manner, we find
X
and it is easy to show that dV/du is of the form
dV _ — mA?) p"m + • • -
du Ag p2m -f • • •
where the degree of the terms not written down is less than 2m in p, and where
the numerator contains no term which does not involve p. As p increases indefi
nitely, the right-hand side approaches — m. Hence R may be chosen so large
that the value of cV/dw, for p = R, is equal to — m + e, where e is less than m
in absolute value. The integral fQ2n( — m + e) du is evidently negative, and
hence the first of the integrals (45) cannot be zero.
EXERCISES
1. At any point of the catenary defined in rectangular coordinates by the
equation
0/5 -
y = - I ea + e
2 \
let us draw the tangent and extend it until it meets the x axis at a point T.
Revolving the whole figure about the x axis, find the difference between the areas
described by the arc AM of the catenary, where A is the vertex of the catenary,
and that described by the tangent M T (I) as a function of the abscissa of the
point M, (2) as a function of the abscissa of the point T.
[Licence, Paris, 1889.]
2. Using the usual system of trirectangular coordinates, let a ruled surface
be formed as follows : The plane zOA revolves about the x axis, while the gen
erating line -D, which lies in this plane, makes with the z axis a constant angle
whose tangent is X and cuts off on OA an intercept OC equal to Xa0, where a
is a given length and 6 is the angle between the two planes zOx and zOA.
VI.Exs.] EXERCISES 293
1) Find the volume of the solid bounded by the ruled surface and the planes
xOy, zOx, and zOA, where the angle 6 between the last two is less than 2n.
2) Find the area of the portion of the surface bounded by the planes xOy,
zOx, zOA.
[Licence, Paris, July, 1882.]
3. Find the volume of the solid bounded by the xy plane, the cylinder
&2X2 _|_ a2y2 — a2&2? anc[ the elliptic paraboloid whose equation in rectangular
coordinates is
2z _ & y*
c p'2 q*
[Licence, Paris, 1882.]
4. Find the area of the curvilinear quadrilateral bounded by the four con-
focal conies of the family
which are determined by giving X the values c2/3, 2c2/3, 4ca/3, 5c2/3, respectively.
[Licence, Besan^on, 1885.]
5. Consider the curve
y = \/2 (sin x — cos x) ,
where x and y are the rectangular coordinates of a point, and where x varies
from 7T/4 to 5?r/4. Find :
1) the area between this curve and the x axis ;
2) the volume of the solid generated by revolving the curve about the x axis ;
3) the lateral area of the same solid.
[Licence, Montpellier, 1898.]
6. In an ordinary rectangular coordinate plane let A and B be any two
points on the y axis, and let AMB be any curve joining A and B which, together
with the line AB, forms the boundary of a region AMBA whose area is a pre-
assigned quantity S. Find the value of the following definite integral taken
over the curve AMB :
— my] dx + [<t>'(y}ex - m]dy,
where m is a constant, and where the function <f>(y), together with its derivative
<t>'(y), is continuous.
[Licence, Nancy, 1895.]
7. By calculating the double integral
> + 00 /» +00
/O
in two different ways, show that, provided that a is not zero,
• + 00 .
sin ax , , ft
ft + 00 ft +00
I e-x!'sinaxdydx
Jo Jo
8. Find the area of the lateral surface of the portion of an ellipsoid of revo
lution or of an hyperboloid of revolution which is bounded by two planes perpen
dicular to the axis of revolution.
294
DOUBLE INTEGRALS
[VI, Exs.
9*. To find the area of an ellipsoid with three unequal axes. Half of the total
area A is given by the double integral
1-
X2-
- C"
dxdy
extended over the interior of the ellipse 62x2 + a2?/2 = a2 ft2. Among the methods
employed to reduce this double integral to elliptic integrals, one of the simplest,
due to Catalan, consists in the transformation used in § 125. Denoting the
integrand of the double integral by v, and letting v vary from 1 to + cc, it is
easy to show that the double integral is equal to the limit, as I becomes infinite,
of the difference
il
7tabl(P-l)
— nab
This expression is an undetermined form ; but we may write
-' --'••>
and hence the limit considered above is readily seen to be
' +00
/.2 /
Ttab
ab
dv
' 1,2 J (V'2_l +^WU2_1+C2\
i \ V a'2'\ &*/ -
10*. If from the center of an ellipsoid whose semiaxes are a, 6, c a perpen
dicular be let fall upon the tangent plane to the ellipsoid, the area of the surface
which is the locus of the foot of the perpendicular is equal to the area of an
ellipsoid whose semiaxes are be/a, ac/b, ab/c.
[WILLIAM ROBERTS, Journal de Liouville, Vol. XI, 1st series, p. 81.]
VI, EXB.] EXERCISES 295
11. Evaluate the double integral of the expression
(* - V)"f(v)
extended over the interior of the triangle bounded by the straight lines y = i0,
y = x, and x = X in two different ways, and thereby establish the formula
' dx £ (x - y)nf(y) dy = I y'^ f(y) dy .
From this result deduce the relation
x- • • CXf(x)dx= l f\x - y)»f(y)dy.
J*9 (n-l)!^x0
In a similar manner derive the formula
Xf(x) dx = — — \ - — f *(x* - 2/2)»
2 . 4 . 6 • • • 2n Jr
and verify these formulae by means of the law for differentiation under the
integral sign.
CHAPTER VII
MULTIPLE INTEGRALS
INTEGRATION OF TOTAL DIFFERENTIALS
I. MULTIPLE INTEGRALS CHANGE OF VARIABLES
143. Triple integrals. Let F(x, y, z) be a function of the three
variables x, y, z which is continuous for all points M, whose rec
tangular coordinates are (cc, y, z), in a finite region of space (£)
bounded by one or more closed surfaces. Let this region be sub
divided into a number of subregions (e^, (e2~), •••, (en), whose vol
umes are vl9 vz, •••, vn, and let (£., 17,-, £;) be the coordinates of any
point m,- of the subregion (e,-). Then the sum
(i)
approaches a limit as the number of the subregions (e.) is increased
indefinitely in such a way that the maximum diameter of each of
them approaches zero. This limit is called the triple integral of
the function F(x, y, z) extended throughout the region (£), and
is represented by the symbol
(2) J J J F(x, y, z} dx dy dz .
The proof that this limit exists is practically a repetition of the
proof given above in the case of double integrals.
Triple integrals arise in various problems of Mechanics, for
instance in finding the mass or the center of gravity of a solid
body. Suppose the region (£) filled with a heterogeneous sub
stance, and let p.(x, y, z) be the density at any point, that is to say,
the limit of the^ratio of the mass inside an infinitesimal sphere about
the point (x, y, «) as center to the volume of the sphere. If ^ and /x2
are the maximum and the minimum value of /t in the subregion (e^),
it is evident that the mass inside that subregion lies between ^v,
and fujv,-; hence it is equal to v,./i(£,., 17.-, £,.), where (£,-, 7;,-, £,-) is a
suitably chosen point of the subregion (e(). It follows that the total
296
VII, §143] INTRODUCTION CHANGE OF VARIABLES 297
mass is equal to the triple integral fffp dx dy dz extended through
out the region (£).
The evaluation of a triple integral may be reduced to the suc
cessive evaluation of three simple integrals. Let us suppose first
that the region (£) is a rectangular parallelepiped bounded by the
six planes x = x0, x = X, y — y0, y — F, z = z0, z = Z. Let (E)
be divided into smaller parallelepipeds by planes parallel to the
three coordinate planes. The volume of one of the latter is
(xf — o^j) (yk — yk_l) (zt — «,_!), and we have to find the limit of
the sum
(3) S =
where the point (£,w, ijikl, £lW) is any point inside the corresponding
parallelepiped. Let us evaluate first that part of S which arises
from the column of elements bounded by the four planes
taking all the points (£ikl, yM, £iw) upon the straight line x = ar.-.j,
y = y/c-i- This column of parallelepipeds gives rise to the sum
(x{ - xi_l)(yt - yt_i)[^(*,-_i, y^i, Ci)(«i - «0) H — ],
and, as in § 123, the £'s may be chosen in such a way that the
quantity inside the bracket will be equal to the simple integral
*(**-!» yt-i) = I F(xi-i> yt-u *)d*-
Jz»
It only remains to find the limit of the sum
But this limit is precisely the double integral
$(x, ?/) dx dy
extended over the rectangle formed by the lines x = x0, x = X,
y = 2/o > y = Y. Hence the triple integral is equal to
I dx I *(x, y)dy,
J*, Jv0
or, replacing &(x, y) by its value,
pX f*Y f*Z
(4) I dx dy F(x,y,z)dz.
•J *,* •Jvn *J *a.
298
MULTIPLE INTEGRALS
[VII, § 144
The meaning of this symbol is perfectly obvious. During the first
integration x and y are to be regarded as constants. The result will be
a function of x and y, which is then to be integrated between the limits
?/0 and F, x being regarded as a constant and y as a variable. The
result of this second integration is a function of x alone, and the last
step is the integration of this function between the limits x0 and X.
There are evidently as many ways of performing this evaluation
as there are permutations on three letters, that is, six. For instance,
the triple integral is equivalent to
r»Z r>X f*Y r>
I dz I dx I F(x, y, z)dy = I
J* J* J« J*a
where *(«) denotes the double integral of F(x, y, z) extended over
the rectangle formed by the lines x = x0, x = X, y = y0, y = Y. We
might rediscover this formula by commencing with the part of the
sum S which arises from the layer of parallelepipeds bounded by the
two planes z = zl_l, z = z(. Choosing the points (£, 77, £) suitably,
the part of S which arises from this layer is
and the rest of the reasoning is similar to that above.
144.
manner
Let us now consider a region of space bounded in any
whatever, and let us divide it into subregions such that any
line parallel to a suitably chosen
fixed line meets the surface which
bounds any subregion in at most
two points. We may evidently
restrict ourselves without loss of
generality to the case in which a
line parallel to the z axis meets
the surface in at most two points.
The points upon the bounding
surface project upon the xy plane
into the points of a region A
bounded by a closed contour C.
To every point (x, y) inside C cor
respond two points on the bound-
ing surface whose coordinates are
FIG. 32
i and
= fa(x> y}- We shall suppose that the functions
are continuous inside C, and that <j>l<<j>t. Let us now
VII, §144] INTRODUCTION CHANGE OF VARIABLES 299
divide the region under consideration by planes parallel to the coor
dinate planes. Some of the subdivisions will be portions of paral
lelepipeds. The part of the sum (1) which arises from the column
of elements bounded by the four planes x = Xf_l} x = x{, y = yk_i,
y = yk is equal, by § 124, to the expression
(xf - «,._!> (yk - yt_,) I J F(x{_ l , yt_lt z) dz + f,k J ,
where the absolute value of eik may be made less than any preassigned
number c by choosing the parallel planes sufficiently near together.
The sum
approaches zero as a limit, and the triple integral in question is
therefore equal to the double integral
<(x, y) dx dy
extended over the region (J) bounded by the contour C, where the
function 3>(z, y) is denned by the equation
r**
= I F(*> y> *)**•
Jz.
If a line parallel to the y axis meets the contour C in at most two
points whose coordinates are y = ^ (x) and y = fa (x), respectively,
while x varies from xl to a;2, the triple integral may also be written
in the form
rxt r«t rzt
(5) / dx \ dy I F(x, y, z)dz.
i/Xj J y^ Jzl
The limits z1 and z2 depend upon both x and y, the limits yt and yz
are functions of x alone, and finally the limits xl and a?2 are constants.
We may invert the order of the integrations as for double inte
grals, but the limits are in general totally different for different
orders of integration.
Note. If ^(x) be the function of x given by the double integral
/•»» rz*
*(«) =| dy I F(x, y, z) dz
Jv, J*,
300 MULTIPLE INTEGRALS [VII, § 145
extended over the section of the given region by a plane parallel to
the yz plane whose abscissa is x, the formula (5) may be written
This is the result we should have obtained by starting with the
layer of subregions bounded by the two planes x = Xf_l} x = xt.
Choosing the points (£, r), £) suitably, this layer contributes to the
total sum the quantity
Example. Let us evaluate the triple integral fffz dx dy dz extended through
out that eighth of the sphere x2 + y* + z2 = fi* which lies in the first octant. If
we integrate first with regard to z, then with regard GO y, and finally with regard
to x, the limits are as follows : x and y being given, z may vary from zero to
V-R2 — x2 — y2 ; x being given, y may vary from zero to V.R2 — x'2 ; and x itself
may vary from zero to B. Hence the integral in question has the value
/*/•/» (*R ~V«2-:r2 />V'ff2-x2-y4
I I I zdxdydz = I dx I dy I zdz,
J J J Jo Jo Jo
whence we find successively
f
Jo
zdz = ±(£2-x2 -y'2),
if
2 Jo
and it merely remains to calculate the definite integral \f0K (R* — x2)^dx, which,
by the substitution x = B cos <f>, takes the form
Hence the value of the given triple integral is, by § 116, x
145. Change of variables. Let
(6)
x = /(?/,, v, w),
y = <£(?*, v, w},
z = $(u, v, w) ,
be formulae of transformation which establish a one-to-one corre
spondence between the points of the region (E~) and those of another
region (^i). We shall think of u, v, and w as the rectangular coor
dinates of a point with respect to another system of rectangular
VII, §145] INTRODUCTION CHANGE OF VARIABLES 301
coordinates, in general different from the first. If F(x, y, «) is a
continuous function throughout the region (£), we shall always have
(7)
mF(x, y, 2) dx dy dz
;>
mF[f(u, v, w),
'i>
D(f,
D(u, v,
dudv dw,
where the two integrals are extended throughout the regions (E)
and (E-i), respectively. This is the formula for change of variables
in triple integrals.
In order to show that the formula (7) always holds, we shall
commence by remarking that if it holds for two or more particular
transformations, it will hold also for the transformation obtained by
carrying out these transformations in succession, by the well-known
properties of the functional determinant (§ 29). If it is applicable
to several regions of space, it is also applicable to the region obtained
by combining them. We shall now proceed to show, as we did for
double integrals, that the formula holds for a transformation which
leaves all but one of the independent variables unchanged, — for
example, for a transformation of the form
(8) x = x1 y = ?/' a = \!/(x' y1 z'}
We shall suppose that the two points M(x, y, z) and M'(x', y',z'} are
referred to the same system of rectangular axes, and that a parallel
to the z axis meets the surface which
bounds the region (E) in at most two
points. The formulae (8) establish a corre
spondence between this surface and another
surface which bounds the region (E'). The
cylinder circumscribed about the two sur
faces with its generators parallel to the
z axis cuts the plane z = 0 along a closed
curve C. Every point m. of the region A
inside the contour C is the projection of
two points m^ and m2 of the first surface,
whose coordinates are z1 and z2, respectively, and also of two
points m[ and m'2 of the second surface, whose coordinates are z[
and z2, respectively. Let us choose the notation in such a way
that zl<z3, and z[<z'2. The formulae (8) transform the point m^
into the point m{, or else into the point m'2. To distinguish the
two cases, we need merely consider the sign of d\f//dz'. If dty/dz' is
C
FIG. 33
302
MULTIPLE INTEGRALS
[VII, § 145
positive, z increases with %', and the points ml and m2 go into the
points m{ and ra^, respectively. On the other hand, if c\(//dz' is
negative, z decreases as 2' increases, and mt and m2 go into ms' and
m{, respectively. In the previous case we shall have
I F(x, y, z)dz = I F^x, y, t(x, y, 2')] ^ dz' ,
*/Zi Jz'i
whereas in the second case
/"**> /*2» 9 /
I F(x, y, z) dz = - I F\x, y, $(x, y, **)] jfc dz' .
Jz^ Jz{
In either case we may write
(9) f *F(x, y, z)dz = f V|>, y, t(x, y, z')]
C/Zj *) ZJ
dz'.
If we now consider the double integrals of the two sides of this
equation over the region A, the double integral of the left-hand side,
II dxdy I F(x, y, z)dz,
J J(A) Jet
is precisely the triple integral fffF(x, y, z) dx dy dz extended through
out the region (E). Likewise, the double integral of the right-hand
side of (9) is equal to the triple integral of
F[x', y',
extended throughout the region (E'~), which readily follows when
x and y are replaced by x' and y', respectively. Hence we have in
this particular case
J J J (E)
F(x, y, z) dx dy dz
dx' di' dz' .
But in this case the determinant D(x, y, z)/D(x', y', z1} reduces to
d(j//dz'. Hence the formula (7) holds for the transformation (8).
Again, the general formula (7) holds for a transformation of the
type
(10) *=/(*', y',*% y = <t>(x',y<,zi),
VII, §145] INTRODUCTION CHANGE OF VARIABLES 303
where the variable z remains unchanged. We shall suppose that
the formulae (10) establish a one-to-one correspondence between
the points of two regions (E) and (£'), and in particular that the
sections R and R' made in (E) and (£'), respectively, by any
plane parallel to the xy plane correspond in a one-to-one manner.
Then by the formulae for transformation of double integrals we
shall have
(11)
I = / / RT fi-r.' i/' ?.'\ A*(r'. -j/ j-'Y f.'l
D(x', y')
, ^ dx'dy'.
The two members of this equation are functions of the variable
z = z' alone. Integrating both sides again between the limits zv
and «2, between which z can vary in the region (£), we find the
formula
Jill F(x> y, z) dx dv dz
J J J(E)
= rrr F^^^,^^,^,^
J J J(E')
D(x',y<)
dx'dy'dz'.
But in this case D(x, y, z)/D(x', y', «') = D(x, y)/D(x', y'~). Hence
the formula (7) holds for the transformation (10) also.
We shall now show that any change of variables whatever
(13) x = f(xl} yl} zj, y = <j>(Xl, yl} zj, z = ^(xlt y1} «j)
may be obtained by a combination of the preceding transformations.
For, let us set x' = xl} y' = ylt z' = z. Then the last equation of
(13) may be written z' = if/(x', y', z^, whence zv = 7r(x', y', z'~).
Hence the equations (13) may be replaced by the six equations
(14) x =/[*', y', TT(X<, y'} «')], y = ftx', V<, ir(x't y', »')], z = z' ,
(15) x' = xi, y' = yi, *' = <j>(xi, yi, «,).
The general formula (7) holds, as we have seen, for each of the
transformations (14) and (15). Hence it holds for the transforma
tion (13) also.
We might have replaced the general transformation (13), as the
reader can easily show, by a sequence of three transformations of
the type (8).
304
MULTIPLE INTEGRALS
[VII, § 146
146. Element of volume. Setting F(x, y, K) = I in the formula (7),
we find
//£***•//£
du dv dw .
The left-hand side of this equation is the volume of the region (£").
Applying the law of the mean to the integral on the right, we find
the relation
(16)
D(u, v,
where Vi is the volume of (^i), and £, rj, £ are the coordinates of some
point in (#1). This formula is exactly analogous to formula (17),
Chapter VI. It shows that the functional determinant is the limit
of the ratio of two corresponding infinitesimal volumes.
If one of the variables u, v, w in (6) be assigned a constant value,
while the others are allowed to vary, we obtain three families of
surfaces, u = const., v = const., w = const., by means of which the
region (£T) may be divided into subregions analogous to the paral-
lelopipeds used above, each of which is bounded by six curved faces.
The volume of one of these subregions bounded by the surfaces
(u), (u + du), (v}, (v + dv), (w), (w + dw} is, by (16),
AF =
. -j- c f c?«< c?v dw ,
D(w, v, w) J
where du, dv, and div are positive increments, and where c is infini
tesimal with du, dv, and dw. The term e du dv dw may be neglected,
as has been explained several times (§ 128). The product
(17)
dV =
D(u, v, w}
is the principal part of the infinitesimal AF, and is called the element
of volume in the system of curvilinear coordinates (u, v, w).
Let c?s2 be the square of the linear element in the same system of
coordinates. Then, from (6),
o/» ^i •/* ^ -P
= ~- - -
dx = ~- du
cu
% ~JL
dw. dy = -r- du
dz = -^- du
cu
=Hldu*+H.tdv*+H3dw'2+2F1dvdw+2F.tdudiv+2F3dudv,
vv Ow
whence, squaring and adding, we find
VII, §14fi] INTRODUCTION CHANGE OF VARIABLES 305
the notation employed being
(19)
(7 IT C OC G OT (JOT O *y
dw du dw ^ du dv
where the symbol AJ means, as usual, that x is to be replaced by y
and z successively and the resulting terms then added.
The formula for dV is easily deduced from this formula for ds*.
For, squaring the functional determinant by the usual rule, we find
dx dy
dz
2
du du
du
ffl
dx d y
d_z
T"t
1 n
dv do
CV
* 3
F,
dx dy
dz
-1 2
dw dw
div
Fl
F,
= M,
whence the element of volume is equal to \M du dv dw.
Let us consider in particular the very important case in which
the coordinate surfaces (u), (?;), (w) form a triply orthogonal system,
that is to say, in which the three surfaces which pass through any
point in space intersect in pairs at right angles. The tangents to
the three curves in which the surfaces intersect in pairs form a tri-
rectangular trihedron. It follows that we must have FI = 0, F2 = 0,
^ = 0; and these conditions are also sufficient. The formulae for
dV and ds* then take the simple forms
a.
(20) ds2 = Hl du* + H2 dv2 + Hs dw2,
dV =
These formulae may also be derived from certain considerations of
infinitesimal geometry. Let us suppose du, dv, and dw very small,
and let us substitute in place of the small subregion defined above a
small parallelopiped with plane faces. Neglecting infinitesimals of
higher order, the three adjacent edges of the parallelopiped may be
taken to be \fn\ dii, ^/If2 dv, and \/^ dw, respectively. The for
mulae (20) express the fact that the linear element and the element of
volume are equal to the diagonal and the volume of this parallelo
piped, respectively. The area ^/H1H2 du dv of one of the faces repre
sents in a similar manner the element of area of the surface (w).
As an example consider the transformation to polar coordinates
(21) x = p sin0cos<£, y = p sin 0
z =
p cos 6,
306
MULTIPLE INTEGRALS
[VII, § 146
where p denotes the distance of the point M(x, y, z) from the origin,
6 the angle between OM and the positive z axis, and <f> the angle
which the projection of OM on the xy plane makes with the positive
x axis. In order to reach all points in space, it is sufficient to let p
vary from zero to + <x>, 0 from zero to TT, and <£ from zero to 2?r.
From (21) we find
(22) ds2 = dp2 + P2 d62 + P2 sin20d<j>2,
whence
(23) dV = p2 sin 6 dp dO d<f> .
These formulae may be derived without any calculation, however.
The three families of surfaces (/a), (0), (<£) are concentric spheres
about the origin, cones of revolution
about the z axis with their vertices
at the origin, and planes through
the z axis, respectively. These
surfaces evidently form a triply
orthogonal system, and the dimen
sions of the elementary subregion
y are seen from the figure to be dp,
p dO, p sin 0 d<j> ; the formulae (22)
and (23) now follow immediately.
To calculate in terms of the va
riables p, 0, and <£ a triple integral
extended throughout a region bounded by a closed surface S, which
contains the origin and which is met in at most one point by a radius
vector through the origin, p should be allowed to vary from zero to R,
where R = /(0, <f>) is the equation of the surface ; 6 from zero to TT ;
and <£ from zero to 2?r. For example, the volume of such a surface is
FIG. 34
V= I d<j> I dO I P2sinOdp.
Jo Jo Jo
The first integration can always be performed, and we may write
R3 sin 0
- c wd+ r
i/O t/0
dQ.
Occasional use is made of cylindrical coordinates r, <a, and z defined
by the equations x = r cos to, y = r sin w, z = z. It is evident that
and
dV = r dw dr dz .
VII, §147] INTRODUCTION CHANGE OF VARIABLES 307
147. Elliptic coordinates. The surfaces represented by the equation
(24)
X — a X — b X — c
-1=0,
where X is a variable parameter and a > b > c> 0, form a family of confocal
conies. Through every point in space there pass three surfaces of this family, —
an ellipsoid, a parted hyperboloid, and an unparted hyperboloid. For the equa
tion (24) always has one root \i which lies between b and c, another root X2
between a and 6, and a third root X3 greater than a. These three roots \i, X2, Xg
are called the elliptic coordinates of the point whose rectangular coordinates are
(x, y, z). Any two surfaces of the family intersect at right angles : if X be given
the values Xi and X2, for instance, in (24), and the resulting equations be sub
tracted, a division by Xi — X2 gives
(25)
- a)
- 6)(X2 - b)
- c)(X2 - c)
= 0,
which shows that the two surfaces (Xi) and (X2) are orthogonal.
In order to obtain x, y, and z as functions of Xi , X2 , X3 , we may note that the
relation
(X - o)(X - 6)(X - c) - x2(X - 6)(X - c) - y2(X - c)(X - a) - z2(X - a)(\ - b)
= (X - Xi)(X - X2)(X - X3)
is identically satisfied. Setting X = a, X = 6, X = c, successively, in this equa
tion, we obtain the values
(X3 - a)(a-X1)(q-X2)
—
(26)
(a — b)(a — c)
(a _ b)(b - c)
(X8 _ C)(X,
(a — c)(b — c)
whence, taking the logarithmic derivatives,
, x
dx = -
2 V Xi - a
X2 — a X3
^)
*-5
+
d\3
— b X2 — b X3 — b
UZ — — I -p ~T I *
2 \Xi — c X2 — c Xs — c/
Forming the sum of the squares, the terms in dXidX2, dX2dX8, dX3dXi must dis
appear by means of (25) and similar relations. Hence the coefficient of dXj is
4 L(Xi - a)2 (\! - 6)2 (X! - c)2
or, replacing x, y, z by their values and simplifying,
(27) 1 (X, - XQft, - XQ
4(X1-a)(X1-6)(X1-c)
308 MULTIPLE INTEGRALS [VII, § 148
The coefficients 3f2 and 3f8 of d\\ and d\g, respectively, may be obtained from
this expression by cyclic permutation of the letters. The element of volume is
therefore -VM i M2 Mz d\i d\2 d\a .
148. Dirichlet's integrals. Consider the triple integral
xpyizr(l — x — y — z)'dxdydz
taken throughout the interior of the tetrahedron formed by the four planes
x = 0, 2/ = 0, z = 0, x + y + z = 1. Let us set
x + y + z = £, y + z = £TJ , z = |r;f ,
where £ , r/, f are three new variables. These formulae may be written in the form
and the inverse transformation is
When x, y, and z are all positive and x + y + z is less than unity, £, 17, and f all
lie between zero and unity. Conversely, if £ , 17, and f all lie between zero and
unity, x, y, and z are all positive and x + y + z is less than unity. The tetra
hedron therefore goes over into a cube.
In order to calculate the functional determinant, let us introduce the auxiliary
transformation X=f, F=£)j, Z = £?;f , which gives x = X — F, y = F — Z ,
z = Z. Hence the functional determinant has the value
D(x, y, z)= D(x, 77, z) D(JT, r, Z) =
D(£, T,, f) D(X, Y, Z) ' J>(|, r,, ^
and the given triple integral becomes
f dg f *i rV+«+r+2(i- O'i7+r+ia- ^ra- f)»df.
t/O «/0 »/0
The integrand is the product of a function of f, a function of 77, and a func
tion of f. Hence the triple integral may be written in the form
f £p + 9 + r + 2(l _ lyfy X C 79 + «- + l(l - T?)?dr, X f
Jo «/o «/o
or, introducing T functions (see (33), p. 280),
l) T(q + r + 2)T(p+l)
T(p + q + r + s + 4) Y(p + q + r + 3) T(q + r + 2)
Canceling the common factors, the value of the given triple integral is finally
found to be
(28) r(p+i)r(g+i)r(r+pr(«+i)
VII, §149] INTRODUCTION CHANGE OF VARIABLES 309
149. Green's theorem.* A formula entirely analogous to (15), § 126, may be
derived for triple integrals. Let us first consider a closed surface S which is
met in at most two points by a parallel to the z axis, and a function R(x, y, z)
which, together with dR/dz, is continuous throughout the interior of this surface.
All the points of the surface S project into points of a region A of the xy plane
which is bounded by a closed contour C. To every point of A inside C corre
spond two points of S whose coordinates are z\ = <f>\ (x, y) and z2 = <j>2 (x, y).
The surface S is thus divided into two distinct portions /Si and 83 . We shall
suppose that z\ is less than z2 .
Let us now consider the triple integral
— dxdydz
dz
taken throughout the region bounded by the closed surface S. A first integra
tion may be performed with regard to z between the limits z\ and z2 (§ 144),
which gives R(x, y, z2) - R(x, y, zt). The given triple integral is therefore
equal to the double integral
J J[Jfc(«, y, z2) - R(x, y, Zi)]dxdy
over the region A. But the double integral f f R(x, y, z2)dxdy is equal to the
surface integral (§ 135)
R(x, y, z)dxdy
taken over the upper side of the surface /S2. Likewise, the double integral of
R(x, y, Zi) with its sign changed is the surface integral
ff R(x,y,z)dxdy
J J(sj
taken over the lower side of <Si . Adding these two integrals, we may write
/ / / aT*6^* =ffs R^ y' Z)dxd2/'
where the surface integral is to be extended over the whole exterior of the sur
face S.
By the methods already used several times in similar cases this formula may
be extended to the case of a region bounded by a surface of any form whatever.
Again, permuting the letters z, y, and z, we obtain the analogous formulae
III ^dxdydz
III -Qdxdydz= C C Q(x, y, z)dzdx.
JJJ** J J<^
* Occasionally called Ostrogradsky's theorem. The theorem of § 126 is sometimes
called Riemann's theorem. But the title Green's theorem is more clearly established
and seems to be the more fitting. See Ency. der Math. Wiss., II, A, 7, b and c. —
TRANS.
310 MULTIPLE INTEGRALS [VII, §150
Adding these three formulae, we finally find the general Green's theorem for
triple integrals :
(29)
= C I P(x, y, z) dy dz + Q(x, y, z) dz dx + R(x, y, z)dxdy,
J J(S)
where the surface integrals are to be taken, as before, over the exterior of the
bounding surface.
If, for example, we set P = x, Q = R = QorQ = y, P — R = Q or R = z,
P = Q = 0, it is evident that the volume of the solid bounded by S is equal to
any one of the surface integrals
(29') CC xdydz, CC ydzdx, CC zdxdy.
J J(S) J J(S) J J(S)
150. Multiple integrals. The purely analytical definitions which have been
given for double and triple integrals may be extended to any number of vari
ables. We shall restrict ourselves to a sketch of the general process.
Let Xi , x2 » • • • t %n be n independent variables. We shall say for brevity
that a system of values x\ , x% , • • • , x°n of these variables represents a point in
space of n dimensions. Any equation F(x\, x2 , • • • , xtl) = 0, whose first member
is a continuous function, will be said to represent a surface; and if F is of the
first degree, the equation will be said to represent a plane. Let us consider the
totality of all points whose coordinates satisfy certain inequalities of the form
(30) ti(xi,X3,-.-,xH)<0, i = l, 2, .-., k.
We shall say that the totality of these points forms a domain D in space of n
dimensions. If for all the points of this domain the absolute value of each of
the coordinates x,- is less than a fixed number, we shall say that the domain D is
finite. If the inequalities which define D are of the form
(31) xJ^X!<x}, x°<;x2<x2, ••-, <<xn^xj,,
we shall call the domain a prismoid, and we shall say that the n positive quan
tities x\ — x°f are the dimensions of this prismoid. Finally, we shall say that a
point of the domain D lies on the frontier of the domain if at least one of the
functions fr in (30) vanishes at that point.
Now let Z) be a finite domain, and let f(x\ , x2 , • • • , xn) be a function which
is continuous in that domain. Suppose D divided into subdomains by planes
parallel to the planes xt- = 0 (t = 1, 2, • • • , n), and consider any one of the pris-
moids determined by these planes which lies entirely inside the domain D.
Let Axi , Ax2 , • • • , Axn be the dimensions of this prismoid, and let £ i , £2 , • • • , £n
be the coordinates of some point of the prismoid. Then the sum
(32) S = S/(fc, £2 ,-•-,£„) AX! Ax2 • • • Ax« ,
formed for all the prismoids which lie entirely inside the domain Z), approaches
a limit I as the number of the prismoids is increased indefinitely in such a way
VII, §150] INTRODUCTION CHANGE OF VARIABLES 311
that all of the dimensions of each of them approach zero. We shall call this
limit I the n-tuple integral of f(x\ , x2, • • • , z,,) taken in the domain D and shall
denote it by the symbol
1 =ff- • "fffai *2, • • •, zn)dzidz2 • • • dxn.
The evaluation of an n-tuple integral may be reduced to the evaluation of
n successive simple integrals. In order to show this in general, we need only
show that if it is true for an (n — l)-tuple integral, it will also be true for an
n-tuple integral. For this purpose let us consider any point (xi , x2 , • • • , xn)
of D. Discarding the variable zn for the moment, the point (x\ , x2 , • • • , xn _ i) evi
dently describes a domain D' in space of (n — 1) dimensions. We shall suppose
that to any point (x1? X2, •••, xn_i) inside of JK there correspond just two
points on the frontier of Z), whose coordinates are (xi, X2, • • •, xn_i; x^) and
(xi, x2, • • • , xn_i ; x^2)), where the coordinates x^ and x^2) are continuous func
tions of the n — 1 variables x\ , x2 , • • • , xn-i inside the domain IX. If this con
dition were not satisfied, we should divide the domain D into domains so small
that the condition would be met by each of the partial domains. Let us now
consider the column of prismoids of the domain D which correspond to the
same point (xi , x2 , • • • , xn-i)- It is easy to show, as we did in the similar case
treated in § 124, that the part of S which arises from this column of prismoids is
r r*w ~\
Ax1Ax2---Axn_i J (i)/(x1,x2, --^x^dXn + e ,
where | e | may be made smaller than any positive number whatever by choos
ing the quantities Ax* sufficiently small. If we now set
r
(33) *(xi, x2, • •-, xn-i)=J (1)
it is clear that the integral I will be equal to the limit of the sum
S*(xi, x2, •••, xn_i)AxiAx2- •• Axn_i,
that is, to the (n — l)-tuple integral
(34) 1 = fff- --J^Xi, x2, • ••, xn-i)dxi---dxn-i,
in the domain ZK. The law having been supposed to hold for an (n — l)-tuple
integral, it is evident, by mathematical induction, that it holds in general.
We might have proceeded differently. Consider the totality of points
(xi , za , • • • , Xn) for which the coordinate xn has a fixed value. Then the
point (xi, x2, • • •, xn_i) describes a domain 8 in space of (n — 1) dimensions,
and it is easy to show that the n-tuple integral I is also equal to the expression
(35) I
where 0(xn) is the («. — l)-tuple integral ///• • -ffdxi • • • dxn-i extended through
out the domain 5. Whatever be the method of carrying out the process, the limits
for the various integrations depend upon the nature of the domain D, and
312 MULTIPLE INTEGRALS [VII, § 150
vary in general for different orders of integration. An exception exists in case
D is a prismoid denned by inequalities of the form
The multiple integral is then of the form
and the order in which the integrations are performed may be permuted in any
way whatever without altering the limits which correspond to each of the
variables.
The formula for change of variables also may be extended to n-tuple integrals.
Let
(36) Xi = 0,-(zi, zg, •••, x'n), t = l, 2, •• -,n,
be formulae of transformation which establish a one-to-one correspondence between
the points (xi , z2 .,-•-, x'n) of a domain jy and the points (xi , x2 , • • • , xn) of a
domain D. Then we shall have
(37)
The proof is similar to that given in analogous cases above. A sketch of the
argument is all that we shall attempt here.
1) If (37) holds for each of two transformations, it also holds for the trans
formation obtained by carrying out the two in succession.
2) Any change of variables may be obtained by combining two transforma
tions of the following types :
(38) Xi = xf, x2 = Xa, •••, xn_i = x;,_i, xn — 0n(xi, xg, • • •, x;,),
(39) zi
3) The formula (37) holds for a transformation of the type (38), since the
given n-tuple integral may be written in the form (34). It also holds for any
transformation of the form (39), by the second form (35) in which the multiple
integral may be written. These conclusions are based on the assumption that
(37) holds for an (n — l)-tuple integral. The usual reasoning by mathematical
induction establishes the formula in general.
As an example let us try to evaluate the definite integral
I = CC- • • f xf'x?2 • • • x"» (1 - xi - x2 ----- xnf dxi dx2--- dxn ,
where ai, cr2, • • • , an, £ are certain positive constants, and the integral is to be
extended throughout the domain D defined by the inequalities
G<X!, 0<z2, •••, 0<xn, xi + x2 + ••• +xn< 1.
The transformation
VII, §151]
TOTAL DIFFERENTIALS
313
carries D into a new domain IX defined by the inequalities
0<fi^l, 0<&<1, •--, 0 <£„<!,
and it is easy to show as in § 148 that the value of the functional determinant is
, X2, • •-, Xn) _ tn-ifc«-2
The new integrand is therefore of the form
^1 + - + «. + «-i^1 + - + «. + -*...^(1_fy(1_&)«1 ...(1 _&,)«-.,
and the given integral may be expressed, as before, in terms of T functions :
(40) I = T(
II. INTEGRATION OF TOTAL DIFFERENTIALS
151. General method. Let P(x, y) and Q(x, y~) be two functions of
the two independent variables x and y. Then the expression
Pdx + Qdy
is not in general the total differential of a single function of the two
variables x and y. For we have seen that the equation
(41) du = Pdx + Qdy
is equivalent to the two distinct equations
(42) £ = *<-,»), £ = «(*,»). .
Differentiating the first of these equations with respect to y and the
second with respect to x, it appears that u(x, y) must satisfy each
of the equations
C2u _ dP(x, y) 82u _ dQ(x, y)
cxdy dy Cydx dx
A necessary condition that a function u(x, y) should exist which
satisfies these requirements is that the equation
(43) £_.!«
Cy dx
should be identically satisfied.
This condition is also sufficient. For there exist an infinite
number of functions u(x, y) for which the first of equations (42)
is satisfied. All these functions are given by the formula
= / P(x,y)dj.
Jxn
314 MULTIPLE INTEGRALS [VII, § 151
where x0 is an arbitrary constant and Fis an arbitrary function of y.
In order that this function u(x, y) should satisfy the equation (41),
it is necessary and sufficient that its partial derivative with respect
to x should be equal to Q(x, y), that is, that the equation
f
Jxn
dp
_<fe+_=^y)
should be satisfied. But by the assumed relation (43) we have
ftx7\T> f*x si f)
I — dx = I -z-dx = Q(x, y) - Q(x0, y) ,
I Cy I CX
whence the preceding relation reduces to
The right-hand side of this equation is independent of x. Hence
there are an infinite number of functions of y which satisfy the
equation, and they are all given by the formula
' = f
•An
Q(«o,
where y0 is an arbitrary value of y, and C is an arbitrary constant.
It follows that there are an infinite number of functions u(x, y)
which satisfy the equation (41). They are all given by the formula
(44) u = f P(x, y} dx + I Q(x0,
Jx0 J'/Q
and differ from each other only by the additive constant C.
Consider, for example, the pair of functions
a; + my y — mx
~ x2 + y* ' x2 + y2 '
which satisfy the condition (43). Setting x0 = 0 and y0 = 1, the
formula for u gives
C*x + my. C'dy ,
u=l , 2dx+ I -^- + C,
Jo x* + y2 Jl y
whence, performing the indicated integrations, we find
1 F x~\x
u = - [log(a;2 + y2)]* + m arc tan - + log y + C,
* L yjo
or, simplifying,
1 x
u = - log(cc2 + y2) -f m arc tan - + C.
* y
VII, §151] TOTAL DIFFERENTIALS 315
The preceding method may be extended to any number of inde
pendent variables. We shall give the reasoning for three variables.
Let P, Q, and R be three functions of x, y, and z. Then the total
differential equation
(45) du = Pdx + Q dy + R dz
is equivalent to the three distinct equations
(46) — = P, — = Q, — = R.
Calculating the three derivatives d^u/dxdy, d2u/dydz, d^u/dzdx in
two different ways, we find the three following equations as neces
sary conditions for the existence of the function u :
d_P_d_Q, 2Q = fL^ <LR = dJL.
dy dx dz dy dx dz
Conversely, let us suppose these equations satisfied. Then, by the
first, there exist an infinite number of functions u(x, y, z) whose
partial derivatives with respect to x and y are equal to P and Q,
respectively, and they are all given by the formula
u = I P(x, y, z)dx + I Q(x0, y, z)dy + Z,
where Z denotes an arbitrary function of z. In order that the deriva
tive du/dz should be equal to R, it is necessary and sufficient that
the equation
should be satisfied. Making use of the relations (47), which were
assumed to hold, this condition reduces to the equation
R(x, y, z) - R(x0, y, z} + R(x0, y, z) - R(x0, y0, z) + — = R(x, y, z} ,
— = R(x0,y0,z).
It follows that an infinite number of functions u(x, y, z) exist
which satisfy the equation (45). They are all given by the formula
f*x pv f*z
(48) u = I P(x, y, z)dx + / Q(x0, y, z)dy+ \ R(x0,y0, z)dz + C,
JxK Jy0 c> za
where x0, yn, za are three arbitrary numerical values, and C is an
arbitrary constant.
316 MULTIPLE INTEGRALS [VII, § 152
152. The integral ^x>y)Pdx + Qdy. The same subject may be
treated from a different point of view, which gives deeper insight
into the question and leads to new results. Let P(x, y) and Q(x, y}
be two functions which, together with their first derivatives, are
continuous in a region A bounded by a single closed contour C.
It may happen that the region A embraces the whole plane, in
which case the contour C would be supposed to have receded to
infinity. The line integral
I Pdx + Qdy
taken along any path D which lies in A will depend in general upon
the path of integration. Let us first try to find the conditions under
which this integral depends only upon the coordinates of the extremi
ties (x0, y0) and (xi} yv) of the path. Let M and N be any two points
of region A, and let L and L' be any two paths which connect these
two points without intersecting each other between the extremities.
Taken together they form a closed contour. In order that the values
of the line integral taken along these two paths L and L' should be
equal, it is evidently necessary and sufficient that the integral taken
around the closed contour formed by the two curves, proceeding
always in the same sense, should be zero. Hence the question at
issue is exactly equivalent to the following : What are the conditions
under which the line integral
f
Pdx + Qdy
taken around any closed contour whatever which lies in the region A
should vanish ?
The answer to this question is an immediate result of Green's
theorem :
Pta + a Ay =//(|f - g
(49)
where C is any closed contour which lies in A, and where the double
integral is to be extended over the whole interior of C. It is clear
that if the functions P and Q satisfy the equation
the line integral on the left will always vanish. This condition is
also necessary. For, if dP/dy — dQ/dx were not identically zero
VII, §152] TOTAL DIFFERENTIALS 317
in the region A, since it is a continuous function, it would surely be
possible to find a region a so small that its sign would be constant
inside of a. But in that case the line integral taken around the
boundary of a would not be zero, by (49).
If the condition (43') is identically satisfied, the values of the
integral taken along two paths L and V between the same two
points M and N are equal provided the two paths do not intersect
between M and N. It is easy to see that the same thing is true
even when the two paths intersect any number of times between M
and N. For in that case it would be necessary only to compare
the values of the integral taken along the paths L and L' with its
value taken along a third path L", which intersects neither of the
preceding except at M and N.
Let us now suppose that one of the extremities of the path of
integration is a fixed point (x0) y0}, while the other extremity is a
variable point (x, y) of A. Then the integral
X(*, V)
Pdx + Qdy
• o- Vo>
taken along an arbitrary path depends only upon the coordinates
(x, y*) of the variable extremity. The partial derivatives of this
function are precisely P(x, y) and Q(x, y). For example, we have
s*(x + A:r, y)
F(x + As, y) = F(x, y} + I P(x, y} dx,
•/(* v)
for we may suppose that the path of integration goes from (x0, y0)
to (x, y), and then from (x, y~) to (x -f Ace, y) along a line parallel to
the x axis, along which dy = 0. Applying the law of the mean, we
may write
F(x + Ace, ?/) — F(
v - '-* - L
= P(x + 0Az, y}, 0<0<1.
u
Taking the limit when Ace approaches zero, this gives Fx = P.
Similarly, Fy = Q. The line integral F(x, y}, therefore, satisfies the
total differential equation (41), and the general integral of this
equation is given by adding to F(x, y) an arbitrary constant.
This new formula is more general than the formula (44) in that
the path of integration is still arbitrary. It is easy to deduce (44)
from the new form. To avoid ambiguity, let (x0, y0) and (a^, yi) be
the coordinates of the two extremities, and let the path of integra
tion be the two straight lines x = x0, y = y^ Along the former,
318
MULTIPLE INTEGRALS
[VII, § 153
x = x0, dx = 0, and y varies from y0 to yt. Along the second,
y = yl} dy = 0, and x varies from x0 to
is equal to
Hence the integral (50)
r«,
I
J;/0 •
which differs from (44) only in notation.
But it might be more advantageous to consider another path of
integration. Let x = /(£), y = <}>(£) be the equations of a curve
joining (x0, y0) and (xlt y^, and let t be supposed to vary con
tinuously from t0 to tl as the point (x, y) describes the curve
between its two extremities. Then we shall have
f lpdx + Qdy = I '
•/<«%, »„> Jf«
where there remains but a single quadrature. If the path be
a straight line, for example, we should set x = x0 + tfa — #„),
y = y0 + t(yl — y,,), and we should let t vary from 0 to 1.
Conversely, if a particular integral ®(x, y) of the equation (41)
be known, the line integral is given by the formula
L
(x. y)
= 3>(x, y) — 3>(x0, yc),
which is analogous to the equation (6) of Chapter IV.
153. Periods. More general cases may be investigated. In the
first place, Green's theorem applies to regions bounded by several
contours. Let us consider for defmiteness a region A bounded by
an exterior contour C and two contours C" and
C" which lie inside the first (Fig. 35). Let P
and Q be two functions which, together with
their first derivatives, are continuous in this
region. (The regions inside the contours C'
and C" should not be considered as parts of
the region A, and no hypothesis whatever is
made regarding P and Q inside these regions.)
Let the contours C' and C" be joined to the contour C by trans
versals ab and cd. We thus obtain a closed contour abmcdndcpbaqa,
or F, which may be described at one stroke. Applying Green's
theorem to the region bounded by this contour, the line integrals
VII, § 153]
TOTAL DIFFERENTIALS
319
which arise from the transversals ab and cd cancel out, since each
of them is described twice in opposite directions. It follows that
f
Pdx + Qdy
•f/Y--
JJ \&;
r/0u
where the line integral is to be taken along the whole boundary of
the region A, i.e. along the three contours C, C', and C", in the senses
indicated by the arrows, respectively, these being such that the
region j\ always lies on the left.
If the functions P and Q satisfy the relation dQ/dx = dP/dy in
the region A, the double integral vanishes, and we may write the
resulting relation in the form
(51) I Pdx + Qdy = I Pdx + Qdy + I Pdx +
J(C) J(C') J(C")
where each of the line integrals is to be taken in the sense desig
nated above.
Let us now return to the region A bounded by a single contour
C, and let P and Q be two functions which satisfy the equation
dP/dy = dQ/dx, and which, together with their first derivatives, are
continuous except at a finite number
of points of A, at which at least one of
the functions P or Q is discontinuous.
We shall suppose for definiteness that
there are three points of discontinuity
a, b, c in A. Let us surround each of
these points by a small circle, and then
join each of these circles to the contour
C by a cross cut (Fig. 36). Then the
integral j Pdx -\-Qdy taken from a fixed
point (x0, T/O) to a variable point (.r, y)
along a curve which does not cross any
of these cuts has a definite value at every point. For the contour C,
the circles and the cuts form a single contour which may be described
at one stroke, just as in the case discussed above. We shall call
such a path direct, and shall denote the value of the line integral
taken along it from M0(x0, y0) to M(x, y) by F(x, y}.
We shall call the path composed of the straight line from M0 to
a point a', whose distance from a is infinitesimal, the circumference
of the circle of radius aa' about a, and the straight line a'M0, a loop-
circuit. The line integral fPdx -f Qdy taken along a loop-circuit
FIG. 36
320 MULTIPLE INTEGRALS [VII, § 153
reduces to the line integral taken along the circumference of the
circle. This latter integral is not zero, in general, if one of the
functions P or Q is infinite at the point a, but it is independent of
the radius of the circle. It is a certain constant ± A, the double
sign corresponding to the two senses in which the circumference
may be described. Similarly, we shall denote by ± B and ± C the
values of the integral taken along loop-circuits drawn about the two
singular points b and c, respectively.
Any path whatever joining M0 and M may now be reduced to a
combination of loop-circuits followed by a direct path from M0 to M.
For example, the path M0mdefM may be reduced to a combination
of the paths M0mdM0, M0deM0, M0efM0, and M0fM. The path
M^mdM^ may then be reduced to a loop-circuit about the singular
point a, and similarly for the other two. Finally, the path M0fM
is equivalent to a direct path. It follows that, whatever be the path
of integration, the value of the line integral will be of the form
(52) F(x, y) = F(x, y) + m A + n B + pC ,
where m, n, and p may be any positive or negative integers. The
quantities A, B, C are called the periods of the line integral. That
integral is evidently a function of the variables x and y which
admits of an infinite number of different determinations, and the
origin of this indetermination is apparent.
Remark. The function F(x, y} is a definitely defined function
in the whole region A when the cuts aa, b/3, cy have been traced.
But it should be noticed that the difference F(m) — F(m') between
the values of the function at two points m and m' which lie on
opposite sides of a cut does not necessarily vanish. For we have
/""' /"»' r*o
A- /: + /+/ ,
i/a; J,,i Jmf
which may be written
But j"^ is zero ; hence
J^-F<X)= A.
It follows that the difference F(ni) — F(m') is constant and equal
to A all along aa. The analogous proposition holds for each of
the cuts.
VII, §154] TOTAL DIFFERENTIALS 321
Example. The line integral
/
i/fll
"
" xdy — ijdx
~f~ y
has a single critical point, the origin. In order to find the corre
sponding period, let us integrate along the circle x2 + y2 = p2.
Along this circle we have
x = p cos «o, y = p sin w, xdy — ydx = p^dto,
whence the period is equal to / "dw = 2?r. It is easy to verify
this, for the integrand is the total differential of arc tan y/x.
154. Common roots of two equations. Let X and Y be two functions of the
variables x and y which, together with their first partial derivatives, are con
tinuous in a region A bounded by a single closed contour C. Then the expres
sion (XdY — YdX)/(X* + Y2) satisfies the condition of integrability, for it is
the derivative of arc tan Y/X. Hence the line integral
(53)
L
X2 +
taken along the contour C in the positive sense vanishes provided the coeffi
cients of dx and dy in the integrand remain continuous inside (7, i.e. if the two
curves X = 0, Y = 0 have no common point inside that contour. But if these
two curves have a certain number of common points a, 6, c, • • • inside C, the value
of the integral will be equal to the sum of the values of the same integral taken
along the circumferences of small circles described about the points a, b, c, • • • as
centers. Let (a, 0) be the coordinates of one of the common points. We shall
suppose that the functional determinant D(X, Y)/D(x, y) is not zero, i.e. that
the two curves X — 0 and Y = 0 are not tangent at the point. Then it is pos
sible to draw about the point (a, /3) as center a circle c whose radius is so small
that the point (JT, Y) describes a small plane region about the point (0, 0)
which is bounded by a contour 7 and which corresponds point for point to the
circle c (§§ 25 and 127).
As the point (x, y) describes the circumference of the circle c in the positive
sense, the point (X, Y) describes the contour y in the positive or in the negative
sense, according as the sign of the functional determinant inside the circle c is
positive or negative. But the definite integral along the circumference of c is
equal to the change in arc tan Y/X in one revolution, that is, ± 2x. Similar
reasoning for all of the roots shows that
'
where P denotes the number of points common to the two curves at which
-^(^, Y)/D(x, y) is positive, and N the number of common points at which the
determinant is negative.
322 MULTIPLE INTEGRALS [VII, § 155
The definite integral on the left is also equal to the variation in arc tan Y/X
in going around c, that is, to the index of the function Y/X as the point (x, y)
describes the contour C. If the functions X and Y are polynomials, and if the
contour C is composed of a finite number of arcs of unicursal curves, we are led
to calculate the index of one or more rational functions, which involves only
elementary operations (§ 77). Moreover, whatever be the functions X and Y,
we can always evaluate the definite integral (54) approximately, with an error
less than 7t, which is all that is necessary, since the right-hand side is always a
multiple of 2it.
The formula (54) does not give the exact number of points common to the
two curves unless the functional determinant has a constant sign inside of C.
Picard's recent work has completed the results of this investigation.*
155. Generalization of the preceding. The results of the preceding paragraphs
may be extended without essential alteration to line integrals in space. Let P,
Q, and -B be three functions which, together with their first partial derivatives,
are continuous in a region (E) of space bounded by a single closed surface S.
Let us seek first to determine the conditions under which the line integral
(55)
•'(•v j/0, z0)
depends only upon the extremities (x0 , yo , Zo) and (x, y, z) of the path of inte
gration. This amounts to inquiring under what conditions the same integral
vanishes when taken along any closed path T. But by Stokes' theorem (§ 136)
the above line integral is equal to the surface integral
CC/
j I (
JJ \
t>p\j /^R f>Q\^ , /5P SR\ 7
— -- }dx dy + ( — -- - I dy dz + ( --- }dz dx
dx dy/ T\a* dz/ ^\cz ex/
extended over a surface S which is bounded by the contour T. In order that
this surface integral should be zero, it is evidently necessary and sufficient that
the equations
8P_BQ ^Q = ^, <3jR = eP
dy dx ' dz dy ' dx dz
should be satisfied. If these conditions are satisfied, U is a function of the vari
ables x, y, and z whose total differential is P dx + Q dy + R dz, and which is single
valued in the region (E). In order to find the value of U at any point, the path
of integration may be chosen arbitrarily.
If the functions P, Q, and E satisfy the equations (56), but at least one of
them becomes infinite at all the points of one or more curves in (E), results
analogous to those of § 153 may be derived.
If, for example, one of the functions P, Q, R becomes infinite at all the points
of a closed curve 7, the integral U will admit a period equal to the value of the
line integral taken along a closed contour which pierces once and only once a
surface <r bounded by 7.
We may also consider questions relating to surface integrals which are exactly
analogous to the questions proposed above for line integrals. Let A, B, and C
be three functions which, together with their first partial derivatives, are
* TraM d' Analyse, Vol. II.
VII, §155] TOTAL DIFFERENTIALS 323
continuous in a region (E) of space bounded by a single closed surface S. Let 2
be a surface inside of (E) bounded by a contour r of any form whatever. Then
the surface integral
(57) I = C f A dy dz + B dz dx + C dx dy
depends in general upon the surface S as well as upon the contour r. In order
that the integral should depend only upon F, it is evidently necessary and suffi
cient that its value when taken over any closed surface in (E) should vanish.
Green's theorem (§ 149) gives at once the conditions under which this is true.
For we know that the given double integral extended over any closed surface is
equal to the triple integral
dA dB dC\,
1 1 \dxdydz
dy dz /
extended throughout the region bounded by the surface. In order that this latter
integral should vanish for any region inside (E), it is evidently necessary that the
functions A, B, and C should satisfy the equation
'dx + Ity + ~dz~ ~
This condition is also sufficient.
Stokes' theorem affords an easy verification of this fact. For if A, J5, and C
are three functions which satisfy the equation (58), it is always possible to deter
mine in an infinite number of ways three other functions P, Q, and R such that
dy dz ~ dz dx dx dy
In the first place, if these equations admit solutions, they admit an infinite
number, for they remain unchanged if P, Q, and R be replaced by
dx ' dy ' dz '
respectively, where X is an arbitrary function of x, y, and z. Again, setting
R = 0, the first two of equations (59) give
P = f *B(x, y, z) dz + <f>(x, y), Q = - C A(x, y, z) dz + $(x, y) ,
where <f>(x. y) and \f/(x, y) are arbitrary functions of x and y. Substituting these
values in the last of equations (59), we find
dA d B\ , d\b d<f> _,.
h — - ) dz H — = C7(«j y, z) ,
dx dy dx dy
~ '0
or, making use of (58),
— = C(x, y, z0) •
dx dy
One of the functions 0 or ^ may still be chosen at random.
The functions P, Q, and R having been determined, the surface integral, by
Stokes' theorem, is equal to the line integral f(r)Pdx + Qdy + Rdz, which
evidently depends only upon the contour F.
324 MULTIPLE INTEGRALS [VII, Exs.
EXERCISES
1. Find the value of the triple integral
(x -y)'2 + 3az- 4a*]dxdydz
extended throughout the region of space defined by the inequalities
x2 + 7/2-az<0, x2 + 2/2 + z2 -2a2<0.
[Licence, Montpellier, 1895.]
2. Find the area of the surface
02X2 + &2y2
and the volume of the solid bounded by the same surface.
3. Investigate the properties of the function
F(X, F, Z) = C dx C dy C /(z, y, z)dz
xo Jo zo
considered as a function of JT, F, and Z. Generalize the results of § 125.
4. Find the volume of the portion of the solid bounded by the surface
(z2 + y2 + z2)8 = 3asxyz
which lies in the first octant.
5. Reduce to a simple integral the multiple integral
C C C a a a
I ••• I x, • xy * • • • x " F(x\ + x2 + • • • + xn) dx\ dx% • • • dxn
J J J
extended throughout the domain D defined by the inequalities
0 < Xi , 0 < Zj , • • • , 0 ^ xn , x\ + x2 + • • • + xn ^ a .
[Proceed as in § 148.]
6. Reduce to a simple integral the multiple integral
extended throughout the domain D defined by the inequalities
7*. Derive the formula
n
iri
f f f--- Cdx1dx«---dxn = — -
JJJ J
VII. Exs.] EXERCISES 325
where the multiple integral is extended throughout the domain D denned by the
inequality
8*. Derive the formula
r27T
'0
C "de C nF(a cos 6 + b sin 0 cos $ + c sin 0 sin 0) sin 6 d<j> - 2n C F(uR) du ,
«/0 Jo •'—I
where a, &, and c are three arbitrary constants, and where B = Va2 + b2 + c2.
[POISSON.]
[First observe that the given double integral is equal to a certain surface inte
gral taken over the surface of the sphere x2 + y2 + z2 = 1. Then take the plane
ax + by + cz = 0 &s the plane of xy in a new system of coordinates.]
9*. Let p = F(6, <f>) be the equation in polar coordinates of a closed surface.
Show that the volume of the solid bounded by the surface is equal to the double
integral
(a) J j ip cos y do-
extended over the whole surface, where da- represents the element of area, and 7
the angle which the radius vector makes with the exterior normal.
10*. Let us consider an ellipsoid whose equation is
and let us define the positions of any point on its surface by the elliptic coordi
nates v and />, that is, by the roots which the above equation would have if /*
were regarded as unknown (cf. § 147). The application of the formulae (29) to
the volume of this ellipsoid leads to the equation
"
r" --- dv m _
Jo
Likewise, the formula (a) gives
>&
M
i/O Jb
- p2) dv
V(62-p2)(C2-p2)(,2-62)(c2-,2) 2
11. Determine the functions P(z, y) and Q(x, y) which, together with their
partial derivatives, are continuous, and for which the line integral
x + a,y + /3)dx + Q(z + a, y + p)dy
taken along any closed contour whatever is independent of the constants a and
/3 and depends only upon the contour itself.
[Licence, Paris, July, 1900.]
326 MULTIPLE INTEGRALS [VII, Exs.
12*. Consider the point transformation defined by the equations
As the point (z', y', z') describes a surface S', the point (x, y, z) describes a sur
face S. Let a, /3, y be the direction angles of the normal to S ; a', 0', -/ the
direction angles of the corresponding normal to the surface S' ; and d<r and d<r'
the corresponding surface elements of the two surfaces. Prove the formula
= ± M ^ y) coscr- + ^ y) cos + ^
r
13*. Derive the formula (16) on page 304 directly.
[The volume V may be expressed by the surface integral
V = I z cos 7 d<r ,
J(S)
and we may then make use of the identity
D(f, *, *) d ( D(/, ») ) 8 j J>(/, 0) ) a
'Y
D(x',y',z') dx'T D(y',z') dy'D(z',x') 3z' ( D(x' , y')
which is easily verified.]
it,
CHAPTER VIII
INFINITE SERIES
I. SERIES OF REAL CONSTANT TERMS
GENERAL PROPERTIES TESTS FOR CONVERGENCE
156. Definitions and general principles. Sequences. The elementary
properties of series are discussed in all texts on College Algebra
and on Elementary Calculus. We shall review rapidly the principal
points of these elementary discussions.
First of all, let us consider an infinite sequence of quantities
(1) SQ, Si, SZ, • • • , Sn ,
in which each quantity has a definite place, the order of precedence
being fixed. Such a sequence is said to be convergent if sn approaches
a limit as the index n becomes infinite. Every sequence which is
not convergent is said to be divergent. This may happen in either
of two ways : sn may finally become and remain larger than any
preassigned quantity, or sn may approach no limit even though it
does not become infinite.
In order that a sequence should be convergent, it is necessary and
sufficient that, corresponding to any preassigned positive number e, a
positive integer n should exist such that the difference sn+p — sn is
less than e in absolute value for any positive integer p.
In the first place, the condition is necessary. For if sn approaches
a limit s as n becomes infinite, a number n always exists for which
each of the differences s — sn, s — sn+l, • • •, s — sn+p, • • • is less than
e/2 in absolute value. It follows that the absolute value of sn+p — sn
will be less than 2 e/2 = c for any value of p.
In order to prove the converse, we shall introduce a very impor
tant idea due to Cauchy. Suppose that the absolute value of each
of the terms of the sequence (1) is less than a positive number N.
Then all the numbers between — N and -f- N may be separated into
two classes as follows. We shall say that a number belongs to the
class A if there exist an infinite number of terms of the sequence (1)
327
328 INFINITE SERIES [VIII, §156
which are greater than the given number. A number belongs to
the class R if there are only a finite number of terms of the
sequence (1) which are greater than the given number. It is
evident that every number between — N and + N belongs to one
of the two classes, and that every number of the class A is less
than any number of the class B. Let S be the upper limit of the
numbers of the class A, which is obviously the same as the lower
limit of the numbers of the class B. Cauchy called this number the
greatest limit (la plus grande des limites) of the terms of the
sequence (1).* This number S should be carefully distinguished
from the upper limit of the terms of the sequence (1) (§ 68). For
instance, for the sequence
11 1
11 2' 3' "' n'
the upper limit of the terms of the sequence is 1, while the greatest
limit is 0.
The name given by Cauchy is readily justified. There always
exist an infinite number of terms of the sequence (1) which lie
between S — « and S + e, however small e be chosen. Let us then
consider a decreasing sequence of positive numbers tl} c2, •••,
en, •••, where the general term ^ approaches zero. To each num
ber €,- of the sequence let us assign a number «. of the sequence (1)
which lies between S — e, and S + e^ We shall thus obtain a
suite of numbers a1? a2, •••, an, • •• belonging to the sequence (1)
which approach S as their limit. On the other hand, it is clear
from the very definition of S that no partial sequence of the kind just
mentioned can be picked out of the sequence (1) which approaches
a limit greater than S. Whenever the sequence is convergent its
limit is evidently the number S itself.
Let us now suppose that the difference sn+p — sn of two terms of
the sequence (1) can be made smaller than any positive' number c
for any value of p by a proper choice of n. Then all the terms of
the sequence past sn lie between slt — e and sn + e. Let S be the
greatest limit of the terms of the sequence. By the reasoning just
given it is possible to pick a partial sequence out of the sequence (1)
which approaches 5 as its limit. Since each term of the partial
sequence, after a certain one, lies between sn — e and sn + c, it is
* Resumes analytiques de Turin, 1833 (Collected Works, 2d series, Vol. X, p. 49).
The definition may be extended to any assemblage of numbers which has an upper
limit.
VIII, Ufi7] CONSTANT TERMS 329
clear that the absolute value of S — sn is at most equal to e. Now
let sm be any term of the sequence (1) whose index m is greater
than n. Then we may write
and the value of the right-hand side is surely less than 2c. Since e
is an arbitrarily preassigned positive number, it follows that the
general term sm approaches S as its limit as the index m increases
indefinitely.
Note. If S is the greatest limit of the terms of the sequence (1),
every number greater than S belongs to the class B, and every num
ber less than S belongs to the class A. The number S itself may
belong to either class.
157. Passage from sequences to series. Given any infinite sequence
the series formed from the terms of this sequence,
(2) U0 + «! + U2 H ----- \- Un -\ ---- ,
is said to be convergent if the sequence of the successive sums
S0 = ?/0, Sl = U0 + 1(1, •••, Sn = U0 + «! + •••+ Un ,
is convergent. Let 5 be the limit of the latter sequence, i.e. the
limit which the sum Sn approaches as n increases indefinitely:
v=n
S = lim S = lim ?/„.
Then S is called the sum of the preceding series, and this relation is
indicated by writing the symbolic equation
S = MO + MI H ----- h un H
A series which is not convergent is said to be divergent.
It is evident that the problem of determining whether the series
is convergent or divergent is equivalent to the problem of determin
ing whether the sequence of the successive sums S0) Slf 52, ••• is
convergent or divergent. Conversely, the sequence
S0) sli S2> " ') sn)
will be convergent or divergent according as the series
SQ + Oi - s0) + («s — *i) H ----- 1- (*„ — a.-i) H ----
330 INFINITE SERIES [VIII, § 157
is convergent or divergent. For the sum Sn of the first n + 1 terms
of this series is precisely equal to the general term sn of the given
sequence. We shall apply this remark frequently.
The series (2) converges or diverges with the series
(3) S + Vn + ••* + *»+€+•••»
obtained by omitting the first p terms of (2). For, if Sn(n > p)
denote the sum of the first n + 1 terms of the series (2), and 2n_p
the sum of the n — p + 1 first terms of the series (3), i.e.
the difference Sn — 2n-P = uo + u\ H ----- h Mp_i is independent of w.
Hence the sum 2n-P approaches a limit if Sn approaches a limit,
and conversely. It follows that in determining whether the series
converges or diverges we may neglect as many of the terms at the
beginning of a series as we wish.
Let S be the sum of a convergent series, Sn the sum of the first
n + 1 terms, and Rn the sum of the series obtained by omitting the
first Ti + 1 terms,
RH = Un + l + «» + 2 H ----- h Un + p -\ ---- .
It is evident that we shall always have
It is not possible, in general, to find the sum S of a convergent
series. If we take the sum 5 of the first n -+- 1 terms as an approxi
mate value of S, the error made is equal to Rn. Since Sn approaches
S as n becomes infinite, the error Rn approaches zero, and hence the
number of terms may always be taken so large — at least theoret
ically — that the error made in replacing S by Sn is less than any
preassigned number. In order to have an idea of the degree of
approximation obtained, it is sufficient to know an upper limit
of Rn. It is evident that the only series which lend themselves
readily to numerical calculation in practice are those for which
the remainder Rn approaches zero rather rapidly.
A number of properties result directly from the definition of con
vergence. We shall content ourselves with stating a few of them.
1) If each of the terms of a given series be multiplied by a constant
k different from zero, the new series obtained will converge or diverge
with the given series; if the given series converges to a sum S, the sum
of the second series is kS.
VIII, §158] CONSTANT TERMS 331
2) If there be given two convergent series
wo + MI + w2 H h MH H ,
i'o + vi + wH 1- vn H ,
whose sums are S and S', respectively, the new series obtained by
adding the given series term by term, namely,
Oo + *'o) + (MX + Vj) + • • • + (M« + «„) + • • • ,
converges, and its sum is S -f- S'. The analogous theorem holds for
the term-by -term addition of p convergent series.
3) The convergence or divergence of a series is not affected if the
values of a finite number of the terms be changed. For such a change
would merely increase or decrease all of the sums Sn after a certain
one by a constant amount.
4) The test for convergence of any infinite sequence, applied to
series, gives Cauchy's general test for convergence : *
In order that a series be convergent it is necessary and sufficient
that, corresponding to any preassigned positive number e, an integer
n should exist, such that the sum of any number of terms what
ever, starting with ull+l, is less than c in absolute value. For
Sn + p — Sn = Un + l + Un + 2 H \~ ?«„+„•
In particular, the general term un+l = Sn+l — Sn must approach
zero as n becomes infinite.
Cauchy's test is absolutely general, but it is often difficult to
apply it in practice. It is essentially a development of the very
notion of a limit. We shall proceed to recall the practical rules most
frequently used for testing series for convergence and divergence.
None of these rules can be applied in all cases, but together they
suffice for the treatment of the majority of cases which actually arise.
158. Series of positive terms. We shall commence by investigating
a very important class of series, — those whose terms are all posi
tive. In such a series the sum Sn increases with n. Hence in
order that the series converge it is sufficient that the sum Sn should
remain less than some fixed number for all values of n. The most
general test for the convergence of such a series is based upon com
parisons of the given series with others previously studied. The
following propositions are fundamental for this process :
* Exercices de Mathtmatlques, 1827. (Collected Works, Vol. VII, 2d series, p. 267.)
332 INFINITE SERIES [VIII, §159
1) If each of the terms of a given series of positive terms is less
than or at most equal to the corresponding term of a known convergent
series of positive terms, the given series is convergent. For the sum
Sn of the first n terms of the given series is evidently less than the
sum S' of the second series. Hence Sn approaches a limit S which
is less than S'.
2) If each of the terms of a given series of positive terms is greater
than or equal to the corresponding term of a known divergent series
of positive terms, the given series diverges. For the sum of the first
n terms of the given series is not less than the sum of the first
n terms of the second series, and hence it increases indefinitely
with n.
We may compare two series also by means of the following
lemma. Let
(U) MO + «! + MJ H h un H ,
(V) v0 + Vl + Va + ... + «, + ...
be two series of positive terms. If the series (£7) converges, and if,
after a certain term, we always have vn+l/vn 5: un+l/un, the series (V)
also converges. If the series (£7) diverges, and if, after a certain
term, we always have un+i/un^vn+}/vn, the series (F) also diverges.
In order to prove the first statement, let us suppose that
vn+i/vn^un+i/un whenever n > 2). Since the convergence of a
series is not affected by multiplying each term by the same con
stant, and since the ratio of two consecutive terms also remains
unchanged, we may suppose that vp < up, and it is evident that we
should have vp + l^up + 1, vp+2^up + 2, etc. Hence the series (F)
must converge. The proof of the second statement is similar.
Given a series of positive terms which is known to converge or
to diverge, we may make use of either set of propositions in order
to determine in a given case whether a second series of positive
terms converges or diverges. For we may compare the terms of
the two series themselves, or we may compare the ratios of two
consecutive terms.
159. Cauchy's test and d'Alembert's test. The simplest series which
can be used for purposes of comparison is a geometrical progression
whose ratio is r. It converges if r < 1, and diverges if r ^ 1. The
comparison of a given series of positive terms with a geometrical
progression leads to the following test, which is due to Cauchy:
VIII, §159] CONSTANT TERMS 333
If the nth root \un of the general term un of a series of positive
terms after a certain term is constantlyjess than a fixed number less
than unity, the series converges. If ~\/un after a certain term is con
stantly greater than unity, the series diverges.
For in the first case ~\/un<k<l, whence un<kn. Hence each
of the terms of the series after a certain one is less than the corre
sponding term of a certain geometrical progression whose ratio is
less than unity. In the second case, on the other hand, ~\/un>l)
whence wn>l. Hence in this case the general term does not
approach zero.
This test is applicable whenever V^ approaches a limit. In
fact, the following proposition may be stated :
If -Vun approaches a limit I as n becomes infinite, the series will
converge if I is less than unity, and it will diverge if I is greater than
unity.
A doubt remains if I = 1, except when ~^fun remains greater than
unity as it approaches unity, in which case the series surely diverges.
Comparing the ratio of two consecutive terms of a given series
of positive terms with the ratio of two consecutive terms of a
geometrical progression, we obtain d'Alembert's test:
If in a given series of positive terms the ratio of any term to the
preceding after a certain term remains less than a fixed number
less than unity, the series converges. If that ratio after a certain
term remains greater than unity, the series diverges.
From this theorem we may deduce the following corollary :
If the ratio un+l/un approaches a limit I as n becomes infinite, the
series converges if I < 1, and diverges ifl>l.
The only doubtful case is that in which 1 = 1; even then, ifun+l/un
remains greater than unity as it approaches unity, the series is divergent.
General commentary. Cauchy's test is more general than d'Alembert's. For
suppose that the terms of a given series, after a certain one, are each less than
the corresponding terms of a decreasing geometrical progression, i.e. that the
general term un is less than Arn for all values of n greater than a fixed integer p,
where A is a certain constant and r is less than unity. Hence Vun < rvl1/", and
the second member of this inequality approaches unity as n becomes infinite.
Hence, denoting by A; a fixed number between r and 1, we shall have after a cer
tain term \/Mn < k. Hence Cauchy's test is applicable in any such case. But it
may happen that the ratio un + \/Un assumes values greater than unity, however
far out in the series we may go. For example, consider the series
1 + r | sin a | + r2 1 sin 2a \ + • • • + rn \ sin na \ -{ ,
'334 INFINITE SERIES [VIII, § 159
where r < 1 and where a is an arbitrary constant. In this case v un = r v | sin net \ < r,
whereas the ratio
sin(n + l)
may assume, in general, an infinite number of values greater than unity as n
increases indefinitely.
Nevertheless, it is advantageous to retain d'Alembert's test, for it is more
convenient in many cases. For instance, for the series
x x2 x3 xn
A H~ ~ i ~ " I ~ ~ ~ ~T~ ' ' ' T
1 1.2 1.2.3 1.2---W,
the ratio of any term to the preceding is x/(n + 1), which approaches zero as n
becomes infinite ; whereas some consideration is necessary to determine inde
pendently what happens to Vun = x/\/l . 2 • • • n as n becomes infinite.
After we have shown by the application of one of the preceding tests that each
of the terms of a given series is less than the corresponding term of a decreasing
geometrical progression A, Ar, Ar2, • • •, Arn, • • •, it is easy to find an upper
limit of the error made when the sum of the first m terms is taken in place of
the sum of the series. For this error is certainly less than the sum of the
geometrical progression
Arm
Arm + Arm+l
When each of the two expressions -\iun and Un + i/uH approaches a limit, the
two limits are necessarily the same. For, let us consider the auxiliary series
. (4) «0 + uix + u2z2 H ----- h unx" H ---- ,
where x is positive. In this series the ratio of any term to the preceding
approaches the limit Ix, where I is the limit of the ratio un + \/un. Hence the
series (4) converges when x < 1/Z, and diverges when x > I /I. Denoting the
limit of -\/Un by *', the expression -\/unxn also approaches a limit Z'x, and
the series (4) converges if x < 1/i', and diverges if x > 1/i'. In order that the
two tests should not give contradictory results, it is evidently necessary that I
and I' should be equal. If, for instance, I were greater than T, the series (4) would
be convergent, by Cauchy's test, for any number x between 1/Z and 1/i', whereas
the same series, for the same value of x, would be divergent by d'Alembert's test.
Still more generally, if Un + \/un approaches a limit Z, >Xun approaches the same
limit.* For suppose that, after a certain term, each of the ratios
Un + \ U,, + 2 U,, + ,,
lies between I — e and I + e, where e is a positive number which may be taken
as small as we please by taking n sufficiently large. Then we shall have
or
*Cauchy, Cours d' Analyse.
Vlll, §160] CONSTANT TERMS 335
As the number p increases indefinitely, while n remains fixed, the two terms on
the extreme right and left of this double inequality approach I -f e and I — e,
respectively. Hence for all values of m greater than a suitably chosen number
we shall have
and, since e is an arbitrarily assigned number, it follows that Vu^ approaches
the number I as its limit.
It should be noted that the converse is not true. Consider, for example, the
sequence
1, a, a6, a"2 6, a262, •••, anbn~l, a"6n, ••-,
where a and b are two different numbers. The ratio of any term to the preced
ing is alternately a and 6, whereas the expression -s/u^ approaches the limit Va6
as n becomes infinite.
The preceding proposition may be employed to determine the limits of cer
tain expressions which occur in undetermined forms. Thus it is evident that
the expression v' 1 . 2 • • • n increases indefinitely with n, since the ratio n \/(n —1)1
increases indefinitely with n. In a similar manner it may be shown that each of
the expressions v'n and v^ogn approaches the limit unity as n becomes infinite.
160. Application of the greatest limit. Cauchy formulated the preceding test
in a more general manner. Let an be the general term of a series of positive
terms. Consider the sequence
1 I I
(5) on, a;, ajj, ••-, a*, •••.
If the terms of this sequence have no upper limit, the general term an will not
approach zero, and the given series will be divergent. If all the terms of the
sequence (5) are less than a fixed number, let w be the greatest limit of the terms
of the sequence.
The series San is convergent if u is less than unity, and divergent if u is greater
than unity.
In order to prove the first part of the theorem, let 1 — a be a number between
w and 1. Then, by the definition of the greatest limit, there exist but a finite
number of terms of the sequence (5) which are greater than 1 — a. It follows
that a positive integer p may be found such that 3/an < 1— a for all values of n
greater than p. Hence the series 2are converges. On the other hand, if « > 1,
let 1 + a be a number between 1 and w. Then there are an infinite number of
terms of the sequence (5) which are greater than 1 + a, and hence there are an
infinite number of values of n for which an is greater than unity. It follows that
the series San is divergent in this case. The case in which w = 1 remains in doubt.
161. Cauchy 's theorem. In case un+l/un and \un both approach
unity without remaining constantly greater than unity, neither
d'Alembert's test nor Cauchy's test enables us to decide whether
the series is convergent or divergent. We must then take as a
comparison series some series which has the same characteristic
336 INFINITE SERIES [VIII, § 161
but which is known to be convergent or divergent. The following
proposition, which Cauchy discovered in studying definite integrals,
often enables us to decide whether a given series is convergent or
divergent when the preceding rules fail.
Let <f>(x) be a function which is positive for values of x greater
than a certain number a, and which constantly decreases as x
increases past x = a, approaching zero as x increases indefinitely.
Then the x axis is an asymptote to the curve y = <f>(%), and the
definite integral
1
<f>(x~)dx
may or may not approach a finite limit as I increases indefinitely.
The series
(6) 4>(a) + 4,(a +!)+••• + <£(> + «) + •••
converges if the preceding integral approaches a limit, and diverges if
it does not.
For, let us consider the set of rectangles whose bases are each
unity and whose altitudes are <f>(a), <f>(a + 1), • • -, <f>(a + n), respec
tively. Since each of these rectangles extends beyond the curve
y — </>(#), the sum of their areas is evidently greater than the area
between the x axis, the curve y = <j>(x), and the two ordinates x = a,
x = a + n, that is,
Xa +
On the other hand, if we consider the rectangles constructed
inside the curve, with a common base equal to unity and with the
altitudes <j>(a +1), <j>(a + 2), • • •, <j>(a -f- n), respectively, the sum of
the areas of these rectangles is evidently less than the area under
the curve, and we may write
Xa + n
+(x)dx.
Hence, if the integral fj'<j>(z) dx approaches a limit L as I increases
indefinitely, the sum <f>(a) -\ ---- + <f>(a + n) always remains less than
<f»(a) -f L. It follows that the sum in question approaches a limit ;
hence the series (6) is convergent. On the other hand, if the inte
gral fa+n<j>(x)dx increases beyond all limit as n increases indefinitely,
the same is true of the sum
<£a -f <t>a + 1 + • • • + <a + ",
VHi, §161] CONSTANT TERMS 337
as is seen from the first of the above inequalities. Hence in this
case the series (6) diverges.
Let us consider, for example, the function <£(ce) = l/x*, where /u,
is positive and a = 1. This function satisfies all the requirements
of the theorem, and the integral // [1/a^] dx approaches a limit as
I increases indefinitely if and only if /u, is greater than unity. It
follows that the series
111 _! + .
1* T 2* 3* n*
converges if p. is greater than unity, and diverges if /x ^ 1.
A.gain, consider the function <£(#) = I/ [x(logo;)'1], where log a;
denotes the natural logarithm, p. is a positive number, and a = 2.
Then, if /* =£ 1, we shall have
/'
c/2
^-[(log7i)1-^-(log2)1-
The second member approaches a limit if /t > 1, and increases
indefinitely with n if /u, < 1. In the particular case when /j. = 1 it
is easy to show in a similar manner that the integral increases
beyond all limit. Hence the series
"t" r« f\ o\u I ' ' ' I /i _\U. I
2 (log 2)* 3 (log 3)* w(logn)'4
converges if /t > 1, and diverges if /u. 51-
More generally the series whose general term is
1
n log 7i log2 n log8 ?i • • • log""1 n(logp ri)*
converges if /t > 1, and diverges if /x ^ 1. In this expression log2 n
denotes log log n, log8 n denotes log log log n, etc. It is understood,
of course, that the integer n is given only values so large that
log n, log2 n, log8 n, •••, logp n are positive. The missing terms in
the series considered are then to be supplied by zeros. The
theorem may be proved easily in a manner similar to the demon
strations given above. If, for instance, /t =£ 1, the function
1
x log x log2 x • • • (logp xY
is the derivative of (log'' a:)1~'i/(l — /A), and this latter function
approaches a finite limit if and only if //. > 1.
338 INFINITE SERIES [vill, § 162
Cauchy's theorem admits of applications of another sort. Let us suppose
that the function </>(x) satisfies the conditions imposed above, and let us con
sider the sum
<j>(n) + 4>(n + 1) + • • • + <f>(n + p) ,
where n andp are two integers which are to be allowed to become infinite. If the
series whose general term is <p(n) is convergent, the preceding sum approaches
zero as a limit, since it is the difference between the two sums Sn + p + i and Sn,
each of which approaches the sum of the series. But if this series is divergent,
no conclusion can be drawn. Keturning to the geometrical interpretation given
above, we find the double inequality
f 0(x)dx<0(n) + 0(n + l) + ••• +<£(n + p)<</>(n).+ f
t/n t/n
* P
Since <f>(ri) approaches zero as n becomes infinite, it is evident that the limit of
the sum in question is the same as that of the definite integral /n"+p0(x)dx,
and this depends upon the manner in which n and p become infinite.
For example, the limit of the sum
1 1 1
n n + 1 n + p
is the same as that of the definite integral f"+P [1/x] dx = log(l + p/n). It is
clear that this integral approaches a limit if and only if the ratio p/n approaches a
limit. If a is the limit of this ratio, the preceding sum approaches log(l + a)
as its limit, as we have already seen in § 49.
Finally, the limit of the sum
Vn Vn + 1 Vn + p
is the same as that of the definite integral
I+P j
f
\J n
— = — 2 ( Vn + p — Vn).
Vx
In order that this expression should approach a limit, it is necessary that the
ratio p/Vn should approach a limit a. Then the preceding expression may be
written in the form
2-
Vn + p + Vn
1
and it is evident that the limit of this expression is a.
162. Logarithmic criteria. Taking the series
11 1
•j^ ' 2M n^
as a comparison series, Cauchy deduced a new test for convergence
which is entirely analogous to that which involves -\/un.
vm, §162] CONSTANT TERMS 339
If after a certain term the expression log(l/wn)/logn- is always
greater than a fixed number which is greater than unity, the series
converges. If after a certain term log(l/wn)/logn is always less
than unity, the series diverges.
If log(l/wn)/log n approaches a limit I as n increases indefinitely,
the series converges if I > 1, and diverges if Ki. The case in
which I = 1 remains in doubt.
In order to prove the first part of the theorem, we will remark
that the inequality
log — > k log n
un •
is equivalent to the inequality
- > nk or un < -T ;
un nk
since k > 1, the series surely converges.
Likewise, if
log— < log w,
Un
we shall have un > 1/n, whence the series surely diverges.
This test enables us to determine whether a given series con
verges or diverges whenever the terms of the series, after a certain
one, are each less, respectively, than the corresponding terms of
the series
where A is a constant factor and p. > 1. For, if
A
Un <—£' '
n
we shall have log ?/„ + //. log n < log A or
, 1
loS^ log A
—. - > /A — •; »
log n log n
and the right-hand side approaches the limit /A as n increases
indefinitely. If K denotes a number between unity and p., we
shall have, after a certain term,
log n
340 INFINITE SERIES [VIII, § 163
Similarly, taking the series
_ _
n(\og n)^ ~r n log w(log2 ri)* '
as comparison series, we obtain an infinite suite of tests for con
vergence which may be obtained mechanically from the preceding
by replacing the expression \og(l/un)/logn by log[l/(/mn)]/log2?i,
then by
log-
nun log n
log8 n
and so forth, in the statement of the preceding tests.* These tests
apply in more and more general cases. Indeed, it is easy to show
that if the convergence or divergence of a series can be established
by means of any one of them, the same will be true of any of those
which follow. It may happen that no matter how far we proceed
with these trial tests, no one of them will enable us to determine
whether the series converges or diverges. Du Bois-Reymond f and
Pringsheim £ have in fact actually given examples of both convergent
and divergent series for which none of these logarithmic tests deter
mines whether the series converge or diverge. This result is of great
theoretical importance, but convergent series of this type evidently
converge very slowly, and it scarcely appears possible that they
should ever have any practical application whatever in problems
which involve numerical calculation. §
163. Raabe's or Duhamel's test. Retaining the same comparison
series, but comparing the ratios of two consecutive terms instead
of comparing the terms themselves, we are led to new tests which
are, to be sure, less general than the preceding, but which are
often easier to apply in practice. For example, consider the series
of positive terms
(7) ?/0 + m + wa H h un -i — ,
* See Bertrand, Traitt de Calcul differential et integral, Vol. I, p. 238; Journal
de Liouville, 1st series, Vol. VII, p. 35.
t Ueber Convergenz von Reihen . . . (Crelle's Journal, Vol. LXXVI, p. 85, 1873).
J Allgemeine Theorie der Divergenz . . . (Mathematische Annalen, Vol. XXXV,
1890).
§In an example of a certain convergent series due to du Bois-Reymond it would
be necessary, according to the author, to take a number of terms equal to the volume
of the earth expressed in cubic millimeters in order to obtain merely half the sum of
the series.
VIII, §163] CONSTANT TERMS 341
in which the ratio un + l/ua approaches unity, remaining constantly
less than unity. Then we may write
«» !+«„
where an approaches zero as n becomes infinite. The comparison of
this ratio with [n/(n +1)]M leads to the following rule, discovered
first by Raabe* and then by Duhamel.f
If after a certain term the product nan is always greater than a
fixed number which is greater than unity, the series converges. If
after a certain term the same product is always less than unity, the
series diverges.
The second part of the theorem follows immediately. For, since
nan < 1 after a certain term, it follows that
1 + acH n+l
and the ratio un + l/un is greater than the ratio of two consecutive
terms of the harmonic series. Hence the series diverges.
In order to prove the first part, let us suppose that after a certain
term we always have nan>k>l. Let p. be a number which lies
between 1 and k, 1 < p. < k. Then the series surely converges if
after a certain term the ratio un + l/un is less than the ratio
[_n/(n + 1)]'1 of two consecutive terms of the series whose general
term is n~*. The necessary condition that this should be true
is that
(8)
or, developing (1 + 1/n)'1 by Taylor's theorem limited to the term
in 1/n2,
l+- + -i< l+an,
W 7T
where Xn always remains less than a fixed number as n becomes
infinite. Simplifying this inequality, we may write it in the form
< nan.
n
* Zeitschrift fur Mathematik und Physik, Vol. X, 1832.
t Journal de Liouville, Vol. IV, 1838.
342 INFINITE SERIES [VIII, §103
The left-hand side of this inequality approaches /u. as its limit as n
becomes infinite. Hence, after a sufficiently large value of n, the
left-hand side will be less than nan, which proves the inequality (8).
It follows that the series is convergent.
If the product nan approaches a limit I as n becomes infinite, we
may apply the preceding rule. The series is convergent if 1>1,
and divergent if I < 1. A doubt exists if I = 1, except when nan
approaches unity remaining constantly less than unity : in that case
the series diverges.
If the product nan approaches unity as its limit, we may compare the ratio
w« + i/^n with the ratio of two consecutive terms of the series
which converges if /*>!> and diverges if M^l- The ratio of two consecutive
terms of the given series may be written in the form
u" 1 ft,
I + - + -
n n
where ft, approaches zero as n becomes infinite. If after a certain term the
product ft, logn is always greater than a fixed number which is greater than unity,
the series converges. If after a certain term the same product is always less than
unity, the series diverges.
In order to prove the first part of the theorem, let us suppose that ft, log n > k > 1.
Let /A be a number between 1 and k. Then the series will surely converge if after
a certain term we have
(9\ u"+l < ~n- [ logn T
un n + 1 Llog(n + l)J
which may be written in the form
logn
or, applying Taylor's theorem to the right-hand side,
1 + - + —
n \ nl ( log n
where XB always remains less than a fixed number as n becomes infinite.
Simplifying this inequality, it becomes
V «
VIII, §lt>;] CONSTANT TERMS 343
The product (n + 1) log (1 + 1/n) approaches unity as n becomes infinite, for it
may be written, by Taylor's theorem, in the form
(10)
where e approaches zero. The right-hand side of the above inequality therefore
approaches fj. as its limit, and the truth of the inequality is established for suffi
ciently large values of n, since the left-hand side is greater than k, which is itself
greater than /^.
The second part of the theorem may be proved by comparing the ratio
ttn + i/w,, with the ratio of two consecutive terms of the series whose general
term is l/(nlogn). For the inequality
«« + 1 > __» l°g n
un n + 1 log(n + 1)
which is to be proved, may be written in the form
1 + -
n n \ n/ 1_ log n
<(* + !) log/1 + --).
V */
The right-hand side approaches unity through values which are greater than
unity, as is seen from the equation (10). The truth of the inequality is there
fore established for sufficiently large values of n, for the left-hand side cannot
exceed unity.
From the above proposition it may be shown, as a corollary, that if the prod
uct (8n log n approaches a limit I as n becomes infinite, the series converges if I > 1,
and diverges if l<l. The case in which I = 1 remains in doubt, unless /3nlogn
is always less than unity. In that case the series surely diverges.
If pn log n approaches unity through values which are greater than unity, we
may write, in like manner,
n n log n
where >„ approaches zero as n becomes infinite. It would then be possible to
prove theorems exactly analogous to the above by considering the product
7,,log2n, and so forth.
Corollary. If in a series of positive terms the ratio of any term to the pre
ceding can be written in the form
. r Hn
— l --- 1
un n n
i + n
where /j. is a positive number, r a constant, and Hn a quantity whose absolute
value remains less than a fixed number as n increases indefinitely, the series con
verges If r is greater than unity, and diverges in all other cases.
344 INFINITE SERIES [VIII, §164
For if we set
we shall have
r - —
nan =
l-r- +
n
and hence lim nan = r. It follows that the series converges if r > 1, and diverges
if r < 1. The only case which remains in doubt is that in which r = 1. In order
to decide this case, let us set
n n
From this we find
log n n + 1 __ log n
-*•*« —
, log n =
and the right-hand side approaches zero as n becomes infinite, no matter how
small the number /* may be. Hence the series diverges.
Suppose, for example, that un + \/un is a rational function of n which ap
proaches unity as n increases indefinitely :
~2 -\ ----
Then, performing the division indicated and stopping with the term in 1/n2, we
may write
Un+l - 1 _L CTl ~ bl -L ^(^
- — i -r -r ., >
un n nz
where 0(n) is a rational function of n which approaches a limit as n becomes
infinite. By the preceding theorem, the necessary and sufficient condition that
the series should converge is that
bi > ai + 1 .
This theorem is due to Gauss, who proved it directly.* It was one of the first
general tests for convergence.
164. Absolute convergence. We shall now proceed to study series
whose terms may be either positive or negative. If after a certain
term all the terms have the same sign, the discussion reduces to
the previous case. Hence we may restrict ourselves to series
which contain an infinite number of positive terms and an infinite
* (Collected Works, Vol. Ill, p. 138.) Disquisitiones generates circa seriem infinitam
a.B
1+ — =» + •••,
l.y
VIII, §164] CONSTANT TERMS 345
number of negative terms. We shall prove first of all the fol
lowing fundamental theorem :
Any series whatever is convergent if the series formed of the abso
lute values of the terms of the given series converges.
Let
(11) UQ + M! H ----- h u» H ----
be a series of positive and negative terms, and let
(12) l\+ L\ + .--+ £/„ + •••
be the series of the absolute values of the terms of the given series,
where Un — \ un . If the series (12) converges, the series (11) like
wise converges. This is a consequence of the general theorem of
§ 157. For we have
and the right-hand side may be made less than any preassigned num
ber by choosing n sufficiently large, for any subsequent choice of p.
Hence the same is true for the left-hand side, and the series (11)
surely converges.
The theorem may also be proved as follows : Let us write
un = (un + Un) - Un,
and then consider the auxiliary series whose general term is un + Un,
(13) (u0 + U0-) + (ui + Ul) + ...+ (un + C7n) + . . . .
Let Sn, S'n, and S'J denote the sums of the first n terms of the series
(11), (12), and (13), respectively. Then we shall have
The series (12) converges by hypothesis. Hence the series (13)
also converges, since none of its terms is negative and its general
term cannot exceed 2Un. It follows that each of the sums S'n and
S'J, and hence also the sum Sn, approaches a limit as n increases
indefinitely. Hence the given series (11) converges. It is evident
that the given series may be thought of as arising from the subtrac
tion of two convergent series of positive terms.
Any series is said to be absolutely convergent if the series of the
absolute values of its terms converges. In such a series the order of
the terms may be changed in any way whatever without altering the
34G INFINITE SERIES [VIII, § KM
sum of the series. Let us first consider a convergent series of posi
tive terms,
(14) a0+aH ----- \-an-\ ---- ,
whose sum is S, and let
(15) b0 + bl + .-. + bn + ..-
be a series whose terms are the same as those of the first series
arranged in a different order, i.e. each term of the series (14) is to
be found somewhere in the series (15), and each term of the series
(15) occurs in the series (14).
Let S'm be the sum of the first m terms of the series (15). Since
all these terms occur in the series (14), it is evident that n may be
chosen so .large that the first m terms of the series (15) are to be
found among the first n terms of the series (14). Hence we shall have
S'm < Sn < S,
which shows that the series (15) converges and that its sum S' does
not exceed S. In a similar manner it is clear that S 5 S'. Hence
S' = S. The same argument shows that if one of the above series
(14) and (15) diverges, the other does also.
The terms of a convergent series of positive terms may also be
grouped together in any manner, that is, we may form a series each
of whose terms is equal to the sum of a certain mimber of terms of
the given series without altering the sum of the series.* Let us first
suppose that consecutive terms are grouped together, and let
(16) A, + Al + Ai + >"+Am + ....
be the new series obtained, where, for example,
Then the sum S'm of the first m terms of the series (16) is equal to
the sum 5V of the first N terms of the given series, where N > m.
As m becomes infinite, N also becomes infinite, and hence S'm also
approaches the limit S.
Combining the two preceding operations, it becomes clear that any
convergent series of positive terms may be replaced by another series
each of whose terms is the sum of a certain number of terms of the
given series taken in any order whatever, without altering the sum of
* It is often said that parentheses may be inserted in a convergent series of positive
terms in any manner whatever without altering the sum of the series. — TRANS.
Vlll, §165] CONSTANT TERMS 347
the series. It is only necessary that each term of the given series
should occur in one and in only one of the groups which form the
terms of the second series.
Any absolutely convergent series may be regarded as the differ
ence of two convergent series of positive terms ; hence the preceding
operations are permissible in any such series. It is evident that an
absolutely convergent series may be treated from the point of view
of numerical calculation as if it were a sum of a finite number of
terms.
165. Conditionally convergent series. A series whose terms do not all
have the same sign may be convergent without being absolutely con
vergent. This fact is brought out clearly by the following theorem
on alternating series, which we shall merely state, assuming that it
is already familiar to the student.*
A series whose terms are alternately positive and negative converges
if the absolute value of each term is less than that of the preceding,
and if, in addition, the absolute value of the terms of the series
diminishes indefinitely as the number of terms increases indefinitely.
For example, the series
i-| + |-* + - + (-l)-; + -
converges. We saw in § 49 that its sum is log 2. The series
of the absolute values of the terms of this series is precisely the
harmonic series, which diverges. A series which converges but
which does not converge absolutely is called a conditionally conver
gent series. The investigations of Cauchy, Lejeune-Dirichlet, and
Riemann have shown clearly the necessity of distinguishing between
absolutely convergent series and conditionally convergent series.
For instance, in a conditionally convergent series it is not always
allowable to change the order of the terms nor to group the terms
together in parentheses in an arbitrary manner. These operations
may alter the sum of such a series, or may change a convergent
series into a divergent series, or vice versa. For example, let us
again consider the convergent series
i-1 + 1-1-» 1 _!_
2,3 4 ^ r2n+l 2n + 2
* It is pointed out in § 1(K> that this theorem is a special case of the theorem proved
there. — TRANS.
348 INFINITE SERIES [VIII, §166
whose sum is evidently equal to the limit of the expression
1 1
!, + 1 2n + 2,
as m becomes infinite. Let us write the terms of this series in another
order, putting two negative terms after each positive term, as follows :
~2~4 + 3~ 6~~8 "*" "*" 2n + 1 ~~ 4r* -f 2 '~ 4» + 4 "*"
It is easy to show from a consideration of the sums S3n, SStl+l, and
S3n+2 that the new series converges. Its sum is the limit of the
expression
yV i _!_ _±
„= o \2n + 1 4:7i + 2 4w + 4
as m becomes infinite. From the identity
2w + 1 4ra + 2 4n + 4 2 \2n + 1 2n + 2,
it is evident that the sum of the second series is half the sum of
the given series.
In general, given a series which is convergent but not absolutely convergent,
it is possible to arrange the terms in such a way that the new series converges
toward any preassigned number A whatever. Let Sp denote the sum of the
first p positive terms of the series, and S'g the sum of the absolute values of the
first q negative terms, taken in such a way that the p positive terms and the q
negative terms constitute the first p + q terms of the series. Then the sum of
the first p + q terms is evidently Sp — S'q. As the two numbers p and q increase
indefinitely, each of the sums Sp and S'q must increase indefinitely, for otherwise
the series would diverge, or else converge absolutely. On the other hand, since
the series is supposed to converge, the general term must approach zero.
We may now form a new series whose sum is A in the following manner :
Let us take positive terms from the given series in the order in which they occur
in it until their sum exceeds A. Let us then add to these, in the order in which
they occur in the given series, negative terms until the total sum is less than A.
Again, beginning with the positive terms where we left off, let us add positive
terms until the total sum is greater than A. We should then return to the
negative terms, and so on. It is clear that the sum of the first n terms of the
new series thus obtained is alternately greater than and then less than A, and
that it differs from A by a quantity which approaches zero as its limit.
166. Abel's test. The following test, due to Abel, enables us to establish the
convergence of certain series for which the preceding tests fail. The proof is
based upon the lemma stated and proved in § 75.
Let
MO + ui + h «„+•••
VIII, §166] CONSTANT TERMS 349
be a series which converges or which is indeterminate (that is, for which the sum
of the first n terms is always less than a fixed number A in absolute value).
Again, let
fQ ) « 1 » ' ' ' ) en ,
be a monotonically decreasing sequence of positive numbers which approach
zero as n becomes infinite. Then the series
(17) foU0 + eiMi + • • • + enWn + . . .
converges under the hypotheses made above.
For by the hypotheses made above it follows that
for any value of n and p. Hence, by the lemma just referred to, we may write
|u,i + ie,, + i + • • • + Un+pen+p\ < 2Aen + i.
Sine? en+i approaches zero as n becomes infinite, n may be chosen so large that
the absolute value of the sum
will be less than any preassigned positive number for all values of p. The
series (17) therefore converges by the general theorem of § 157.
When the series u0 + MI + • • • + un + • • • reduces to the series
1 - 1 + 1 - 1 + 1 -!••-,
whose terms are alternately + 1 and — 1, the theorem of this article reduces to
the theorem stated in § 165 with regard to alternating series.
As an example under the general theorem consider the series
sin 6 + sin 2 0 + sin 3 6 + • • • + sin n 0 + • • • ,
which is convergent or indeterminate. For if sin 6 = 0, every term of the series
is zero, while if sin 0 ^ 0, the sum of the first n terms, by a formula of Trigo
nometry, is equal to the expression
. nft
sin —
2 . /n + 1
sin (
.9 \ 2
sin -
2
which is less than | I/sin (6/2) \ in absolute value. It follows that the series
sin 6 sin 2 0 sin n 6
I i T I • • • -f- -j-
converges for all values of 6. It may be shown in a similar manner that the
series
cos 6 cos 2 6 cos n 6
12 n
converges for all values of 0 except 0 — 2krt.
h
350 INFINITE SERIES [VIII, § 167
Corollary. Restricting ourselves to convergent series, we may state a more
general theorem. Let
MO + MI H ----- f- un + • • •
be a convergent series, and let
be any monotonically increasing or decreasing sequence of positive numbers
which approach a limit k different from zero as n increases indefinitely. Then
the series
(18) e0«o + ei«i H ----- h «»«„ H ----
also converges.
For definiteness let us suppose that the e's always increase. Then we may
write
e0 = k — a0 , ei = k — a-i , • • • , en = k — an , • • • ,
where the numbers a0 , a\ , • • • , an , • • • form a sequence of decreasing positive
numbers which approach zero as n becomes infinite. It follows that the two
series
ku0 + kui + • • • + kun + • • • ,
both converge, and therefore the series (18) also converges.
II. SERIES OF COMPLEX TERMS MULTIPLE SERIES
167. Definitions. In this section we shall deal with certain gen
eralizations of the idea of an infinite series.
Let
(19) u0 + u, + i,z + • • • + un + • • •
be a series whose terms are imaginary quantities:
Such a series is said to be convergent if the two series formed of
the real parts of the successive terms and of the coefficients of the
imaginary parts, respectively, both converge:
(20) a0 + a, + a2 -f . • • + an + • • • = S',
(21) &. + &1+6.+- -+^ + --- = ^".
Let S' and S" be the sums of the series (20) and (21), respectively.
Then the quantity S = S' + is" is called the sum of the series (19).
It is evident that S is, as before, the limit of the sum Sn of the first
n terms of the given series as n becomes infinite. It is evident
that a series of complex terms is essentially only a combination of
two series of real terms.
VIII, §168] COMPLEX TERMS MULTIPLE SERIES 351
When the series of absolute values of the terms of the series (19)
converges, each of the series (20) and (21) evidently converges abso
lutely, for \an < vX + °l and IVI =
In this case the series (19) is said to be absolutely convergent. The
sum of such a series is not altered by a change in the order of the
terms, nor by grouping the terms together in any way.
Conversely, if each of the series (20) and (21) converges absolutely,
the series (22) converges absolutely, for y a* + b\ ;> | an | + | bn \ .
Corresponding to every test for the convergence of a series of
positive terms there exists a test for the absolute convergence of
any series whatever, real or imaginary. Thus, if the absolute value
of the ratio of two consecutive terms of a series \un + l/un\, after a cer
tain term, is less than a fixed number less than unity, the series con
verges absolutely. For, let £7, = | u, \ . Then, since | un+l/un \ < k < 1
after a certain term, we shall have also
U „ rl _ ,
which shows that the series of absolute values
U0 + Ul + • • • + Un + • • •
converges. If \un+l/un[ approaches a limit I as n becomes infinite,
the series converges if I < 1, and diverges if I > 1. The first half is
self-evident. In the second case the general term un does not
approach zero, and consequently the series (20) and (21) cannot
both be convergent. The case I = 1 remains in doubt.
More generally, if to be the greatest limit of Vt7n as n becomes infinite, the
series (19) converges if w<l, and diverges if w>l. For in the latter case the
modulus of the general term does not approach zero (see § 161). The case in
which w = 1 remains in doubt — the series may be absolutely convergent, simply
convergent, or divergent.
168. Multiplication of series. Let
(23) u» + Ul + uz + ••• + "„+••-,
(24) r0 + Vl + v, + • • • + vn + • • •
be any two series whatever. Let us multiply terms of the first
series by terms of the second in all possible ways, and then group
352 INFINITE SERIES [VIII, § 168
together all the products utVj for which the sum i+j of the sub
scripts is the same ; we obtain in this way a new series
(25) \ U°V° + (H°VI + UlV°) + (M°*'2 + u*vi + W2W») ^
( +(u<>vn + ulvn_l-{ hwBVo)H •
If each of the series (23) and (24) is absolutely convergent, the
series (25) converges, and its sum is the product of the sums of the
two given series. This theorem, which is due to Cauchy, was gener
alized by Mertens,* who showed that it still holds if only one of the
series (23) and (24) is absolutely convergent and the other is merely
convergent.
Let us suppose for definiteness that the series (23) converges
absolutely, and let wn be the general term of the series (25):
Wn = «0^n + MlW»-l H 1- UnV0.
The proposition will be proved if we can show that each of the
differences
^0 + Wl H 1" WSn ~ (U0 + % -{ 1- «„) (l'0 + Vl H h Vn) ,
^o + wl -\ + w2n+l - (u0 + ul-\ + «n + i)0o + *i H + vn + 1)
approaches zero as n becomes infinite. Since the proof is the same
in each case, we shall consider the first difference only. Arranging
it according to the u's, it becomes
S = U0(vn + l -\ h V2n) + HI (Wn + 1 + • • • + V2n_,) H h M.,-1 t'. + i
+ wn+i(v0H -f v»_i)+ un + a(v0-\ h vn_2)-\ \-u2nv0.
Since the series (23) converges absolutely, the sum U0 + Ul H f- Un
is less than a fixed positive number A for all values of n. Like
wise, since the series (24) converges, the absolute value of the sum
vo + Vi + • • • + vn is less than a fixed positive number B. Moreover,
corresponding to any preassigned positive number e a number m
exists such that
A + B
c
for any value of p whatever, provided that n > m. Having so chosen
n that all these inequalities are satisfied, an upper limit of the quan
tity 1 8 is given by replacing u0, ul} uz, • • ., u2n by U0, U1} U2, • • •, L\nJ
* Crelle's Journal, Vol. LXXIX.
VIII, §169] COMPLEX TERMS MULTIPLE SERIES 353
respectively, vn + l + vn + 2 -\ ----- \- vn + p by (./(A + B), and finally each
of the expressions v0 + vt + • • • + vn _ l , v0 + • • • + vn _ 2 , • • • , v0 by B.
This gives
1 8 1 < u° + Ul + ' " + Un ' l
or
A + B A + B
whence, finally, 1 8 1 < e. Hence the difference 8 actually does approach
zero as n becomes infinite.
169. Double series. Consider a rectangular network which is lim
ited upward and to the left, but which extends indefinitely down
ward and to the right. The network will contain an infinite number
of vertical columns, which we shall number from left to right from
0 to + oo . It will also contain an infinite number of horizontal
rows, which we shall number from the top downward from 0 to + oo .
Let us now suppose that to each of the rectangles of the network a
certain quantity is assigned and written in the corresponding rec
tangle. Let aik be the quantity which lies in the ith row and in the
kih column. Then we shall have an array of the form
(26)
We shall first suppose that each of the elements of this array is real
and positive.
Now let an infinite sequence of curves Clt C2, • • •, Cn, • • • be drawn
across this array as follows : 1) Any one of them forms with the two
straight lines which bound the array a closed curve which entirely
surrounds the preceding one ; 2) The distance from any fixed point
to any point of the curve CB, which is otherwise entirely arbitrary,
becomes infinite with n. Let St be the sum of the elements of the
array which lie entirely inside the closed curve composed of C,- and
354 INFINITE SERIES [VIII, § 169
the two straight lines which bound the array. If Sn approaches a
limit S as n becomes infinite, we shall say that the double series
•\-~r- +-r>
(27)
converges, and that its sum is S. In order to justify this definition,
it is necessary to show that the limit 5 is independent of the form
of the curves C. Let C{, C"z, •••, C'm, ••• be another set of curves
which recede indefinitely, and let S{ be the sum of the elements
inside the closed curve formed by C- and the two boundaries. If m
be assigned any fixed value, n can always be so chosen that the
curve Cn lies entirely outside of C'm. Hence S'm< Sn, and therefore
S'm ^ S, for any value of m. Since S'm increases steadily with m, it
must approach a limit S' < 5 as m becomes infinite. In the same
way it follows that S < S'. Hence S' = S.
For example, the curve C. may be chosen as the two lines which
form with the boundaries of the array a square whose side increases
indefinitely with i, or as a straight line equally inclined to the two
boundaries. The corresponding sums are, respectively, the following :
«00 + fan) + «11 + «Ol)H ----- Ka»0 + an\ H ----- h «„» + «»-!, n H ----- H «0n) >
If either of these sums approaches a limit as n becomes infinite, the
other will also, and the two limits are equal.
The array may also be added by rows or by columns. For, sup
pose that the double series (27) converges, and let its sum be 5. It
is evident that the sum of any finite number of elements of the series
cannot exceed 5. It follows that each of the series formed of the
elements in a single row
(28) %• + •*! + — + <*. + —•, i = 0, 1, 2, • • •,
converges, for the sum of the first n + 1 terms a;o + an + • • • + a,n
cannot exceed S and increases steadily with n. Let o-, be the sum of
the series formed of the elements in the ith row. Then the new series
(29) o-0 + o-, + - • • + o-, + • • •
surely converges. For, let us consider the sum of the terms of the
array 2«tt for which i^p, k^r. This sum cannot exceed S, and
increases steadily with r for any fixed value of p; hence it
approaches a limit as r becomes infinite, and that limit is equal to
(30) <rn + o-, -f- • • • + tr
VIII, §1(50] COMPLEX TERMS MULTIPLE SERIES 355
for any fixed value of p. It follows that <r0 + o^ + • • • + vp cannot
exceed 5 and increases steadily with p. Consequently the series (29)
converges, and its sum 2 is less than or equal to S. Conversely, if
each of the series (28) converges, and the series (29) converges to a
sum 2, it is evident that the sum of any finite number of elements
of the array (26) cannot exceed 2. Hence S 5= 2, and consequently
2 = s.
The argument just given for the series formed from the elements
in individual rows evidently holds equally well for the series formed
from the elements in individual columns. The sum of a convergent
double series whose elements are all positive may be evaluated by
rows, by columns, or by means of curves of any form which recede
indefinitely. In particular, if the series converges when added by rows,
it will surely converge when added by columns, and the sum will be the
same. A number of theorems proved for simple series of positive
terms may be extended to double series of positive elements. For
example : if each of the elements of a double series of positive elements
is less, respectively, than the corresponding elements of a knoivn con
vergent double series, the first series is also convergent; and so forth.
A double series of positive terms which is not convergent is said
to be divergent. The sum of the elements of the corresponding
array which lie inside any closed curve increases beyond all limit
as the curve recedes indefinitely in every direction.
Let us now consider an array whose elements are not all positive.
It is evident that it is unnecessary to consider the cases in which
all the elements are negative, or in which only a finite number of
elements are either positive or negative, since each of these cases
reduces immediately to the preceding case. We shall therefore sup
pose that there are an infinite number of positive elements and an
infinite number of negative elements in the array. Let alk be the
general term of this array T. If the array 7\ of positive elements,
each of which is the absolute value | aik\ of the corresponding element
in T, converges, the array T is said to be absolutely convergent. Such
an array has all of the essential properties of a convergent array of
positive elements.
In order to prove this, let us consider two auxiliary arrays T'
and T", defined as follows. The array T is formed from the array T
by replacing each negative element by a zero, retaining the positive
elements as they stand. Likewise, the array T" is obtained from
the array T by replacing each positive element by a zero and chang
ing the sign of each negative element. Each of the arrays T' and T"
356 INFINITE SERIES [VIII, § 169
converges whenever the array 7\ converges, for each element of T',
for example, is less than the corresponding element of 7^. The sum
of the terms of the series T which lie inside any closed curve is
equal to the difference between the sum of the terms of T which
lie inside the same curve and the sum of the terms of T" which
lie inside it. Since the two latter sums each approach limits as
the curve recedes indefinitely in all directions, the first sum also
approaches a limit, and that limit is independent of the form of
the boundary curve. This limit is called the sum of the array T.
The argument given above for arrays of positive elements shows
that the same sum will be obtained by evaluating the array T by
rows or by columns. It is now clear that an array whose elements
are indiscriminately positive and negative, if it converges absolutely,
may be treated as if it were a convergent array of positive terms.
But it is essential that the series 7\ of positive terms be shown to
be convergent.
If the array TI diverges, at least one of the arrays T' and T" diverges. If
only one of them, T' for example, diverges, the other T" being convergent, the
sum of the elements of the array T which lie inside a closed curve C becomes
infinite as the curve recedes indefinitely in all directions, irrespective of the
form of the curve. If both arrays 7" and T" diverge, the above reasoning
shows only one thing, — that the sum of the elements of the array T inside
a closed curve C is equal to the difference between two sums, each of which
increases indefinitely as the curve C recedes indefinitely in all directions. It
may happen that the sum of the elements of T inside C approach different
limits according to the form of the curves C and the manner in which they
recede, that is to say, according to the relative rate at which the number of
positive terms and the number of negative terms in the sum are made to increase.
The sum may even become infinite or approach no limit whatever for certain
methods of recession. As a particular case, the sum obtained on evaluating by
rows may be entirely different from that obtained on evaluating by columns if
the array is not absolutely convergent.
The following example is due to Arndt.* Let us consider the array
(31)
1 /1\ 1 /2\ 1 /2\ 1 /3
p
2\2/ SW 3\3/ 4\4/ ' 'A P I P +
l^V-^-V 1/2V-1/3V I/^liV--1 X n X'
2V2/ 3V3/'3V3/ 4\4/ '' P\ P / P
I /l\" 1 /2\n 1 /2\" 1 /3\"
2\2/ 3\3/ 3V3/ 4\4/ p\ p / p +
* Grunert's Archiv, Vol. XI, p. 319.
VIII, §169] COMPLEX TERMS MULTIPLE SERIES 357
which contains an infinite number of positire and an infinite number of negative
elements. Each of the series formed from the elements in a single row or from
those in a single column converges. The sum of the series formed from the
terms in the nth row is evidently
2\2
Hence, evaluating the array (31) by rows, the result obtained is equal to the
sum of the convergent series
2a + 28 + '" + 2Ml + '"'
which is 1/2. On the other hand, the series formed from the elements in the
(p — l)th column, that is,
converges, and its sum is
jj-1 p -1 1 1
p p+ 1 p(p +1) p + 1 p
Hence, evaluating the array (31) by columns, the result obtained is equal to the
sum of the convergent series
3
which is — 1/2.
This example shows clearly that a double series should not be used in a
calculation unless it is absolutely convergent.
We shall also meet with double series whose elements are complex
quantities. If the elements of the array (26) are complex, two other
arrays 7" and T" may be formed where each element of T' is the
real part of the corresponding element of T and each element of T"
is the coefficient of i in the corresponding element of T. If the
array 7\ of absolute values of the elements of T, each of whose
elements is the absolute value of the corresponding element of T,
converges, each of the arrays T' and T" converges absolutely, and
the given array T is said to be absolutely convergent. The sum of
the elements of the array which lie inside a variable closed curve
approaches a limit as the curve recedes indefinitely in all directions.
This limit is independent of the form of the variable curve, and it
is called the sum of the given array. The sum of any absolutely
convergent array may also be evaluated by rows or by columns.
358 INFINITE SERIES [VIII, § 170
170. An absolutely convergent double series may be replaced by a simple
series formed from the same elements. It will be sufficient to show that the
rectangles of the network (26) can be numbered in such a way that each rec
tangle has a definite number, without exception, different from that of any other
rectangle. In other words, we need merely show that the sequence of natural
numbers
(32) 0, 1, 2, -.., n, -.-,
and the assemblage of all pairs of positive integers (i, fc), where i^O, k>0, can
be paired off in such a way that one and only one number of the sequence (32)
will correspond to any given pair (i, k), and conversely, no number n corresponds
to more than one of the pairs (t, k). Let us write the pairs (i, k) in order as
follows :
(0,0), (1,0), (0,1), (2,0), (1,1), (0,2), -..,
where, in general, all those pairs for which i + k = n are written down after
those for which i + k < n have all been written down, the order in which those
of any one set are written being the same as that of the values of i for the various
pairs beginning with (n, 0) and going to (0, n). It is evident that any pair (i, k)
will be preceded by only a, finite number of other pairs. Hence each pair will
have a distinct number when the sequence just written down is counted off
according to the natural numbers.
Suppose that the elements of the absolutely convergent double series SSaa- are
written down in the order just determined. Then we shall have an ordinary series
(33) doo + «io + «oi + «2o + «ii + floa + • • • + a«o + a»-i,i + • • •
whose terms coincide with the elements of the given double series. This simple
series evidently converges absolutely, and its sum is equal to the sum of the given
double series. It is clear that the method we have employed is not the only pos
sible method of transforming the given double series into a simple series, since
the order of the terms of the series (33) can be altered at pleasure. Conversely,
any absolutely convergent simple series can be transformed into a double series
in an infinite variety of ways, and that process constitutes a powerful instrument
in the proof of certain identities.*
It is evident that the concept of double series is not essentially different from
that of simple series. In studying absolutely convergent series we found that
the order of the terms could be altered at will, and that any finite number of
terms could be replaced by their sum without altering the sum of the series.
An attempt to generalize this property leads very naturally to the introduction
of double series.
171. Multiple series. The notion of double series may be generalized.
In the first place we may consider a series of elements amn with two
subscripts ra and n, each of which may vary from — oo to + oo .
The elements of such a series may be arranged in the rectangles of
a rectangular network which extends indefinitely in all directions ;
*Tanuery, Introduction a hi theorie desf auctions d'une variable, p. G7.
VIII, §172] COMPLEX TERMS MULTIPLE SERIES 359
it is evident that it may be divided into four double series of the
type we have just studied.
A more important generalization is the following. Let us consider
a series of elements of the type «„„,,„„. ..,mp, where the subscripts
tK.1} m2 , • • • , mp may take on any values from 0 to -f oo , or from — GO
to + oo, but may be restricted by certain inequalities. Although no
such convenient geometrical form as that used above is available
when the number of subscripts exceeds three, a slight consideration
shows that the theorems proved for double series admit of immediate
generalization to multiple series of any order p. Let us first sup
pose that all the elements «MI, „,,,..., mp are real and positive. Let Sl
be the sum of a certain number of elements of the given series, S2
the sum of <S\ and a certain number of terms previously neglected,
Sa the sum of S2 and further terms, and so on, the successive sums
S1} S2, ••-, Sn, ••• being formed in such a way that any particular
element of the given series occurs in all the sums past a certain one.
If Sn approaches a limit S as n becomes infinite, the given series
is said to be convergent, and S is called its sum. As in the case of
double series, this limit is independent of the way in which the
successive sums are formed.
If the elements of the given multiple series have different signs
or are complex quantities, the series will still surely converge if the
series of absolute values of the terms of the given series converges.
172. Generalization of Cauchy's theorem. The following theorem,
which is a generalization of Cauchy's theorem (§ 161), enables us to
determine in many cases whether a given multiple series is conver
gent or divergent. Let/(.x, y) be a function of the two variables x
and y which is positive for all points (x, y) outside a certain closed
curve T, and which steadily diminishes in value as the point (x, y)
recedes from the origin.* Let us consider the value of the double
integralJJ"/^, y} dx dij extended over the ring-shaped region between
T and a variable curve C outside T, which we shall allow to recede
indefinitely in all directions ; and let us compare it with the double
series 2/(wi, n), where the subscripts m and n may assume any posi
tive or negative integral values for which the point (m, n) lies out
side the fixed curve T. Then the double series converges if the double
integral approaches a limit, and conversely.
* All that is necessary for the present proof is that/to, l/i)>/(a;2) 7/2) whenever
x\>X2 ond y\>y^ outside T. It is easy to adapt the proof to still more general
hypotheses. — TRANS.
360 INFINITE SERIES [VIH,§173
The lines x = 0, x = ± 1, x = ± 2, • • • and y = 0, y = ± 1, y = ± 2, • • •
divide the region between F and C into squares or portions of squares.
Selecting from the double series the term which corresponds to that
corner of each of these squares which is farthest from the origin, it
is evident that the sum 2/(w, n) of these terms will be less than the
value of the double integral ///(#, y) dx dy extended over the region
between F and C. If the double integral approaches a limit as C
recedes indefinitely in all directions, it follows that the sum of any
number of terms of the series whatever is always less than a fixed
number ; hence the series converges. Similarly, if the double series
converges, the value of the double integral taken over any finite
region is always less than a fixed number ; hence the integral
approaches a limit. The theorem may be extended to multiple
series of any order p, with suitable hypotheses ; in that case the
integral of comparison is a multiple integral of order p.
As an example consider the double series whose general term is
l/(?n2 + n2)M, where the subscripts m and n may assume all integral
values from — oo to + °o except the values m = n — 0. This series
converges for p. > 1, and diverges for /x^l. For the double integral
dx dy
extended over the region of the plane outside any circle whose
center is the origin has a definite value if ^ > 1 and becomes
infinite if p.<l (§133).
More generally the multiple series whose general term is
1
(m\ + m\ + -.. + miy'
where the set of values ml = mz = • • • = mp = 0 is excluded, con
verges if 2n > j9.*
III. SERIES OF VARIABLE TERMS UNIFORM CONVERGENCE
173. Definition of uniform convergence. A series of the form
(35) u0(x) + KJ (*)+... + un(x) + ••-,
whose terms are continuous functions of a variable x in an inter
val (a, i), and which converges for every value of x belonging to
that interval, does not necessarily represent a continuous function,
*More general theorems are to be found in Jordan's C'ours d' Analyse, Vol. I, p. 163.
VIII, §173] VARIABLE TERMS 361
as we might be tempted to believe. In order to prove the fact we
need only consider the series studied in § 4 :
1 + x2 (1 + a;2)2 (1 + a;2)"
which satisfies the above conditions, but whose sum is discontinuous
for x — 0. Since a large number of the functions which occur in
mathematics are defined by series, it has been found necessary to
study the properties of functions given in the form of a series. The
first question which arises is precisely that of determining whether
or not the sum of a given series is a continuous function of the
variable. Although no general solution of this problem is known,
its study has led to the development of the very important notion
of uniform, convergence.
A series of the type (35), each of whose terms is a function of x
which is defined in an interval (a, 6), is said to be uniformly con
vergent in that interval if it converges for every value of x between
a and b, and if, corresponding to any arbitrarily preassigned positive
number c, a positive integer N, independent of x, can be found such
that the absolute value of the remainder Rn of the given series
Rn = Un + l («) + MH + S (*) H ----- H ^n+p(x) -\ ----
is less than e for every value of n^N and for every value of x
which lies in the interval (a, &).
The latter condition is essential in this definition. For any pre
assigned value of x for which the series converges it is apparent
from the very definition of convergence that, corresponding to any
positive number e, a number N can be found which will satisfy
the condition in question. But, in order that the series should con
verge uniformly, it is necessary further that the same number N
should satisfy this condition, no matter what value of x be selected
in the interval (a, b). The following examples show that such is not
always the case. Thus in the series considered just above we have
The series in question is not uniformly convergent in the inter
val (0, 1). For, in order that it should be, it would be necessary
(though not sufficient) that a number N exist, such that
1
362 INFINITE SERIES [VIII, § 17.5
for all values of x in the interval (0, 1), or, what amounts to the
same thing, that
Whatever be the values of N and e, there always exist, however,
positive values of x which do not satisfy this inequality, since the
right-hand side is greater than unity.
Again, consider the series denned by the equations
= S - S
n_lf =
The sum of the first n terms of this series is evidently Sn (#), which
approaches zero as n increases indefinitely. The series is therefore
convergent, and the remainder Rn (cc) is equal to — nxe~"x*. In order
that the series should be uniformly convergent in the interval (0, 1),
it would be necessary and sufficient that, corresponding to any arbi
trarily preassigned positive number e, a positive integer N exist such
that for all values of n > N
But, if x be replaced by 1/w, the left-hand side of this inequality is
equal to e~l/n, which is greater than 1/e whenever n > 1. Since e
may be chosen less than 1/e, it follows that the given series is not
uniformly convergent.
The importance of uniformly convergent series rests upon the
following property:
The sum of a series whose terms are continuous functions of a
variable x in an interval (a, i) and which converges uniformly in that
interval, is itself a continuous function of x in the same interval.
Let XQ be a value of x between a and b, and let x0 -f h be a value
in the neighborhood of XQ which also lies between a and b. Let n
be chosen so large that the remainder
fl»(«) = un + l(x) + un+z(x) H ----
is less than e/3 in absolute value for all values of x in the interval
(a, b), where e is an arbitrarily preassigned positive number. Let/(cc)
be the sum of the given convergent series. Then we may write
where <}>(x) denotes the sum of the first n -f 1 terms,
«K*) = MuO*0 + M, 00 H — + «„(»).
VIII, § 17.5] VARIABLE TERMS
Subtracting the two equalities
363
f(x0 + It) = <f>(xu + A) + Rtt(x0 + A),
we find
f(x0 + h) -/(*„) = [>(x0 + A) - <£(*„)] + /^(^u + A) - TJ.^o).
The number « was so chosen that we have
O O
On the other hand, since each of the terms of the series is a continu
ous function of x, <£(o:) is itself a continuous function of x. Hence
a positive number 77 may be found such that
whenever | h is less than rj. It follows that we shall have, a fortiori,
whenever \h is less than rj. This shows that f(x) is continuous
for x = x0.
Note. It would seem at first very difficult to determine whether
or not a given series is uniformly convergent in a given interval.
The following theorem enables us to show in many cases that a
given series converges uniformly.
Let
(36) MO (a:) + MI (x) + - . - + «„(*) + • • •
be a series each of whose terms is a continuous function of x in an
interval (a, b), and let
(37) Jf. + Jtfi + ... + *; + ...
be a convergent series whose terms are positive constants. Then,
tf I un | ^ Mn for aH values of x in the interval (a, b) and for all
values of n, the first series (36) converges uniformly in the interval
considered.
For it is evident that we shall have
364 INFINITE SERIES [VIII, § 174
for all values of x between a and b. If N be chosen so large that
the remainder Rn of the second series is less than e for all values
of n greater than N, we shall also have
whenever n is greater than N, for all values of x in the interval (a, b).
For example, the series
M0 + Mi sin x + Mjj sin 2x -\ ----- \- Masmnx -\ ---- ,
where M0, Ml} Mz, ••• have the same meaning as above, converges
uniformly in any interval whatever.
174. Integration and differentiation of series.
Any series of continuous functions which converges uniformly in an
interval (a, b) may be integrated term by term, provided the limits of
integration are finite and lie in the interval (a, b*).
Let x0 and xv be any two values of x which lie between a and b,
and let N be a positive integer such that | RB(ar)| < e for all values
of x in the interval (a, b) whenever n > N. Let f(x) be the sum of
the series
and let us set
/•*i. rxi rxi rx\ rx*
Dn — \ f(x) dx — I u0dx — I Uidx ----- I undx = I Rndx.
Jxt Jr0 tA0 Jx0 Jx0
The absolute value of Dn is less than c x^ — x0 1 whenever n^
Hence Dn approaches zero as n increases indefinitely, and we have
the equation
r*i r*i rx* r*1
I f(x}dx=\ u0(x)dx+l Ui(x)dx-\ ----- hi uH(x)dx -\ ---- .
J*t Jft J*« ^x*
Considering XQ as fixed and Xj as variable, we obtain a series
I uQ(x)dx-\ ---- +lun(x')dx-\ ----
J*t J*o
which converges uniformly in the interval (a, b) and represents a
continuous function whose derivative is f(x).
VIII, §174] VARIABLE TERMS 365
Conversely, any convergent series may be differentiated term by term
if the resulting series converges uniformly.*
For, let
f(x) = M0 (a;) + Ul(x) + • • • + un(x) -\ ----
be a series which converges in the interval (a, ft). Let us suppose
that the series whose terms are the derivatives of the terms of the
given series, respectively, converges uniformly in the same interval,
and let <£(z) denote the sum of the new series
Integrating this series term by term between two limits x0 and x,
each of which lies between a and b, we find
/'
JxK
= [u0 (a:) - MO(*O)] + [MJ (a;) -
*SJC0
or
This shows that <f>(x) is the derivative of /(x) .
Examples. 1) The integral
— dx
x
cannot be expressed by means of a finite number of elementary
functions. Let us write it as follows :
/ex C dx C ex — 1 C ex — 1
— dx = I (-/ dx = log x -f I dx.
x J x J x J x
The last integral may be developed in a series which holds for all
values of x. For we have
and this series converges uniformly in the interval from — R to + R,
no matter how large R be taken, since the absolute value of any
* It is assumed in the proof also that each term of the new series is a continuous
function. The theorem is true, however, in general. — TRANS.
366 INFINITE SERIES [VIII, § 174
term of the series is less than the corresponding term of the con
vergent series
It follows that the series obtained by term-by-term integration
OC OC^ 3Cn
F^=l+I + 2Y72 + ' ' + nl.2...n + '
converges for any value of x and represents a function whose deriva
tive is (tf — !)/£.
2) The perimeter of an ellipse whose major axis is 2a and whose eccentricity
is e is equal, by § 1 12, to the definite integral
= 4c f
«/o
The product e2sin2 <j> lies between 0 and e2(< 1). Hence the radical is equal to the
sum of the series given by the binomial theorem
Vl — e2 sin2 <f> = 1 -- e2sin20 -- e4sin*^> — • ••
'
2 . 4 . 6 • • • 2n
The series on the right converges uniformly, for the absolute value of each of
its terms is less than the corresponding term of the convergent series obtained
by setting sin 0 = 1. Hence the series may be integrated term by term; and
since, by § 116,
C\ . .,. 1.8.6...(2n-l) it
sin^"0a* — -- 1
Jo 2 . 4 . 6 • • • 2n 2
we shall have
f Vl-e2sinV<Z0 = -Sl- - e2 - — e* - — eft ----
Jo 2 ( 4 64 256
3.6...(2n-3)-|2
2.4.6--.2n JV
If the eccentricity e is small, a very good approximation to the exact value of the
integral is obtained by computing a few terms.
Similarly, we may develop the integral
f Vl- e2sin20drf>
Jo
in a series for any value of the upper limit <f>.
Finally, the development of Legendre's complete integral of the first kind
leads to the formula
Vlll, §174] VARIABLE TERMS 367
The definition of uniform convergence may be extended to series
whose terms are functions of several independent variables. For
example, let
u0(x, y) + % (x, y) H h un(x, y}-\
be a series whose terms are functions of two independent variables x
and ij, and let us suppose that this series converges whenever the
point (x, y} lies in a region R bounded by a closed contour C.
The series is said to be uniformly convergent in the region R if,
corresponding to every positive number e, an integer N can be found
such that the absolute value of the remainder Rn is less than e
whenever n is equal to or greater than N, for every point (x, y)
inside the contour C. It can be shown as above that the sum of
such a series is a continuous function of the two variables x and
y in this region, provided the terms of the series are all continu
ous in R.
The theorem on term-by-term integration also may be generalized.
If each of the terms of the series is continuous in R and if f(x, y)
denotes the sum of the series, we shall have
I I f(x> y}dxdy =11 u9(x, y) dx dy + I I ul(x, y)dxdy -\
+ 11 un(x,y}dxdy-\ ,
where each of the double integrals is extended over the whole inte
rior of any contour inside of the region R.
Again, let us consider a double series whose elements are functions
of one or more variables and which converges absolutely for all sets
of values of those variables inside of a certain domain D. Let the
elements of the series be arranged in the ordinary rectangular array,
and let Rc denote the sum of the double series outside any closed
curve C drawn in the plane of the array. Then the given double
series is said to converge uniformly in the domain D if correspond
ing to any preassigned number c, a closed curve K, not dependent
on the values of the variables, can be drawn such that | Rc < e for
any curve C whatever lying outside of K and for any set of values
of the variables inside the domain D.
It is evident that the preceding definitions and theorems may be
extended without difficulty to a multiple series of any order whose
elements are functions of any number of variables.
368 INFINITE SERIES [VIII, § 175
Note. If a series does not converge uniformly, it is not always allowable to
integrate it term by term. For example, let us set
SH (x) = nxe-»*?, S0(x) = 0, ua (x) = Sn - Sn^ . n = 1, 2, • • • .
The series whose general term is un (x) converges, and its sum is zero, since Sn (x)
approaches zero as n becomes infinite. Hence we may write
f(x) = 0 = 1*1 (z) + MS (x) + • • • + u, (x) + • • • ,
whence J f(x) dx = 0. On the other hand, if we integrate the series term by
term between the limits zero and unity, we obtain a new series for which the
sum of the first n terms is
which approaches 1/2 as its limit as n becomes infinite.
175. Application to differentiation under the integral sign. The proof
of the formula for differentiation under the integral sign given in
§ 97 is based essentially upon the supposition that the limits x0
and X are finite. If X is infinite, the formula does not always hold.
Let us consider, for example, the integral
f + " s'max
F(a) = I dx , a > 0 .
i/O X
This integral does not depend on a, for if we make the substitu
tion y = ax it becomes
y
If we tried to apply the ordinary formula for differentiation to F(a~),
we should find
F'(o-) = I cos ax dx
Jo
This is surely incorrect, for the left-hand side is zero, while the
right-hand side has no definite value.
Sufficient conditions may be found for the application of the
ordinary formula for differentiation, even when one of the limits
is infinite, by connecting the subject with the study of series. Let
us first consider the integral
p + »
f(x)dx,
which we shall suppose to have a determinate value (§ 90). Let
au a2, • ••, an, • • • be an infinite increasing sequence of numbers, all
Vni, §175] VARIABLE TERMS 369
greater than a0, where an becomes infinite with n. If we set
/"«2 ra»+i
Ja^ Jan
the series
converges and its sum is j*" f(x) dx, for the sum Sn of the first n terms
is equal to fannf(x) dx.
It should ° be noticed that the converse is not always true.
If, for example, we set
f(x) = cosx, a0 — 0, «! = TT, ••-, an = mr, ••-,
we shall have
Un = I cos x dx = 0 .
U nit
Hence the series converges, whereas the integral fo cosxdx ap
proaches no limit whatever as I becomes infinite.
Now let f(x, a) be a function of the two variables x and a which
is continuous whenever x is equal to or greater than a0 and a lies
in an interval (a0> «i)- If the integral Ja f(x, a) dx approaches a
limit as I becomes infinite, for any value" of a, that limit is a
function of a,
r+x
(a)= f(x,a)dx,
i/a.
which may be replaced, as we have just shown, by the sum of a
convergent series whose terms are continuous functions of a :
U0 (a) = I \f(x, a) dx , U, (a) = / /(ar, a) rfx ,
A, ^«t
This function F(a) is continuous whenever the series converges uni
formly. By analogy we shall say that the integral f*mf(xt a) dx
converges uniformly in the interval (a0, aj if, corresponding to any
preassigned positive quantity e, a number N independent of a can
be found such that | ft + "f(x, a)dx < e whenever I > N, for any value
of a which lies in the interval (a0, a^* If the integral converges
* See W. F. OSGOOD, Annals of Mathematics, 2d series, Vol. Ill (1902), p. 129. —
TRANS.
370 INFINITE SERIES [vm,§i75
uniformly, the series will also. For if an be taken greater than N,
we shall have
\R.\ =
r+*
I f(x, cr) dx
Jan
hence the function F(a) is continuous in this case throughout the
interval (a0, a^.
Let us now suppose that the derivative df/da is a continuous
function of x and a when x ^ a0 and a0 < a < a^ , that the integral
da
has a finite value for every value of a in the interval (a0, a^, and
that the integral converges uniformly in that interval. The integral
in question may be replaced by the sum of the series
— dx = F0 (a-) + F! (tr) + • • • + Fn (a)
where
The new series converges uniformly, and its terms are equal to the
corresponding terms of the preceding series. Hence, by the theorem
proved above for the differentiation of series, we may write
In other words, the formula for differentiation under the integral sign
still holds, provided that the integral on the right converges uniformly.
The formula for integration under the integral sign (§ 123) also
may be extended to the case in which one of the limits becomes
infinite. Let f(x, a) be a continuous function of the two variables
x and a, for x > a0, a0 < a < at . If the integral // V(*, «) dx is uni
formly convergent in the interval (a0, a^, we shall have
r + * r«^ f«, /> + »
(A) / dx I f(x,a}da=\ da f(x,a)dx.
«A| ^«0 J«0 J«0
To prove this, let us first select a number I > a0 ; then we shall
have
Cl ra* rai rl
(B) / dx \ f(x, a)da= I da f(x, a)dx.
^ua J*f J«0 Ja0
VIII, § 176]
VARIABLE TERMS 371
As I increases indefinitely the right-hand side of this equation
approaches the double integral
. + ao
f(x, a)dx,
for the difference between these two double integrals is equal to
ai r+x
da I /(cc, <x)dx.
Jl
Suppose N chosen so large that the absolute value of the integral
/ + :0/(x, a)dx is less than c whenever / is greater than N, for any
value of a in the interval (a0, a^. Then the absolute value of the
difference in question will be less than c a1 — a0 \ , and therefore it
will approach zero as I increases indefinitely. Hence the left-hand
side of the equation (B) also approaches a limit as I becomes infi
nite, and this limit is represented by the symbol
X+°0 /»«!
dx I f(x, a) da.
J*t
This gives the formula (A) which was to be proved.*
176. Examples. 1) Let us return to the integral of § 91 :
/•» + »
F(a}- I er"* — - dx,
Jo x
where a is positive. The integral
/• + '-«
— I er ax sin x dx ,
Jo
* The formula for differentiation may be deduced easily from the formula (A). For,
suppose that the two functions f(x, a) and fa (x, a) are continuous for a0 < a < en ,
x ;> a0 ; that the two integrals F(a) = fa+ */(*, «) dx and *(«) = fa* *S« (x> °^ dx have
finite values ; and that the latter converges uniformly in the interval (a0, «i). From
the formula (A), if a lies in the interval (a0> <*!)• we have
fduC "/«<*t«)d*= f dx C fu(x,u)du,
J"o J"o Jmt Jao
where for distinctness a has heen replaced by u under the integral sign. But this
formula may be written in the form
f "*(!*) du = ( + */(*, a)dx- C+ ™f(x,
•Jao Jao Jao
whence, taking the derivative of each side with respect to a, we find
372 INFINITE SERIES [VIII, § 176
obtained by differentiating under the integral sign with respect to a, converges
uniformly for all values of a greater than an arbitrary positive number k. For
we have
/» +<=0 ~ +00 1
I e-axsmxdx<l e~axdx = — eral .
Ji Ji a
and hence the absolute value of the integral on the left will be less than e for all
values of a greater than k, if I > N, where N is chosen so large that keky > 1/e
It follows that
•*- \ (X) — — I Q sin £ ctj& ,
Jo
The indefinite integral was calculated in § 119 and gives
~e-<*x(cosx + a sinx)~| + °° —1
F'(a) = I"
1 + a2 Jo 1 + a2
whence we find
F(a) = C - arc tan a ,
and the constant C may be determined by noting that the definite integral F(a)
approaches zero as a becomes infinite. Hence C = x/2, and we finally find the
formula
r + »
,. sin x , 1
e~ ax dx = arc tan — .
J + »
Sll
e~ax —
',
This formula is established only for positive values of n-, but we saw in § 91 that
the left-hand side is the sum of an alternating series whose remainder Rn is always
less than 1/n. Hence the series converges uniformly, and the integral is a con
tinuous function of a, even for a = 0. As a approaches zero we shall have in
the limit
f+ao
Jo
2) If in the formula
(39)
2
rv-<fc=^
Jo 2
of § 134 we set x = yVa, where a is positive, we find
(40) r + V
t/o
and it is easy to show that all the integrals derived from this one by successive
differentiations with respect to the parameter a converge uniformly, provided
that a is always greater than a certain positive constant A;. From the preceding
formula we may deduce the values of a whole series of integrals :
22
(41)
VIII, Exs.] EXERCISES 373
By combining these an infinite number of other integrals may be evaluated.
We have, for example,
/^ -4- oo
- I t-**W**
Jo ! • 2
All the integrals on the right have been evaluated above, and we find
/- + « 1 \n (2/3)2 Vn a-i
1 ^0032^ = -^--^ — — + ...
+ /_iy, (2fl2" ^* 1.8.6.. .(8n-l)8
7 1.2.3.--2n 2 2"
or, simplifying,
/• + *> 1 Iff _ 5!
(42) I e~av*cos2pydy = ~ •%/- e « .
^o 2 \a
EXERCISES
1. Derive the formula
; — [Z" (logZ)><] = 1 + Si lOgZ + — ^- (logz)2 + h " (logz)" ,
1 . 2 • • • n uz" 1.2 1 . 2 • • • n
where Sp denotes the sum of the products of the first n natural numbers taken p
at a time. r,, n
[MURPHY.]
[Start with the formula
1.2
and differentiate n times with respect to z.]
2. Calculate the value of the definite integral
(log»)» + ..1
1 . 2 • • • n
j»
by means of the formula for differentiation under the integral sign.
3. Derive the formula
/
r+* -+ ^
=1 e &dx =
Jo
[First show that dl/da = - 21.]
374 INFINITE SERIES rviil,
4. Derive the formula
J_a_ *? da i— ,.
e « — = = VTre-2*
Va
by making use of the preceding exercise.
5. From the relation
derive the formula
CHAPTER IX
POWER SERIES TRIGONOMETRIC SERIES
In this chapter we shall study two particularly important classes
of series — power series and trigonometric series. Although we shall
speak of real variables only, the arguments used in the study of
power series are applicable without change to the case where the
variables are complex quantities, by simply substituting the expres
sion modulus or absolute value (of a complex variable) for the expres
sion absolute value (of a real variable).*
I. POWER SERIES OF A SINGLE VARIABLE
177. Interval of convergence. Let us first consider a series of the form
(1) A0 + AiX + A2X* + • • • -f AHX» + ••-,
where the coefficients A0, A1} A2, ••• are all positive, and where
the independent variable A' is assigned only positive values. It is
evident that each of the terms increases with A'. Hence, if the
series converges for any particular value of A, say Xl} it converges
a fortiori for any value of A less than Xl. Conversely, if the series
diverges for the value A2, it surely diverges for any value of A
greater than A2. We shall distinguish the following cases.
1) The series (1) may converge for any value of A whatever.
Such is the case, for example, for the series
Y A2 A"
1"fI + lT2 + >" + l72T^ + ""
2) The series (1) may diverge for any value of A except A = 0
The following series, for example, has this property :
1 + A + 1 . 2A2 H +1.2.3-.- nXn H .
3) Finally, let us suppose that the series converges for certain
values of A and diverges for other values. Let Ax be a value of A
for which it converges, and let A2 be a value for which it diverges.
* See Vol. II, §§ 2GG-275. — TKAKS.
375
376 SPECIAL SERIES [IX, § 177
From the remark made above, it follows that Xl is less than X2. The
series converges if A'<A'j, and it diverges if AT>A'2. The only
uncertainty is about the values of A between Xl and X2. But all
the values of A' for which the series converges are less than X2, and
hence they have an upper limit, which we shall call R. Since all the
values of X for which the series diverges are greater than any value
of A' for which it converges, the number R is also the lower limit of
the values of X for which the series diverges. Hence the series (1)
diverges for all values of X greater than R, and converges for all values
of X less than R. It may either converge or diverge when X = R.
For example, the series
converges if A' < 1, and diverges if A' ^ 1. In this case R =1.
This third case may be said to include the other two by suppos
ing that R may be zero or may become infinite.
Let us now consider a power series, i.e. a series of the form
(2) a0 -f a1x + azx2 + • • • + anxa + • • •,
where the coefficients a, and the variable x may have any real values
whatever. From now on we shall set A{ = |a,-|, X = \x\. Then the
series (1) is the series of absolute values of the terms of the series (2).
Let R be the number defined above for the series (1). Then the
series (2) evidently converges absolutely for any value of x between
- R and + R, by the very definition of the number R. It remains
to be shown that the series (2) diverges for any value of x whose
absolute value exceeds R. This follows immediately from a funda
mental theorem due to Abel : *
If the series (2) converges for any particular value x0 , it converges
absolutely for any values of x whose absolute value is less than \x0\.
In order to prove this theorem, let us suppose that the series (2)
converges for x = x0, and let M be a positive number greater than
the absolute value of any term of the series for that value of x.
Then we shall have, for any value of n,
Alt x0n<M,
and we may write
/ \ _ /
<M/X
, m in (m — 1)
* Recherche sur la sene 1 H x -\
1 . 2t
IX, §177] POWER SERIES 377
It follows that the series (1) converges whenever X<|cc0|, which
proves the theorem.
In other words, if the series (2) converges for x — x0, the series (1)
of absolute values converges whenever X is less than | x0 1 . Hence
|cc0 cannot exceed R, for R was supposed to be the upper limit of
the values of X for which the series (1) converges.
To sum up, given a power series (2) whose coefficients may have
either sign, there exists a positive number R which has the follow
ing properties : The series (2) converges absolutely for any value of x
between — R and -f R, and diverges for any value of x whose absolute
value exceeds R. The interval (— R, + R) is called the interval of
convergence. This interval extends from — oo to + cc in the case in
which R is conceived to have become infinite, and reduces to the
origin if R = 0. The latter case will be neglected in what follows.
The preceding demonstration gives us no information about what
happens when x = R or x = — R. The series (2) may be absolutely
convergent, simply convergent, or divergent. For example, R = 1
for each of the three series
for the ratio of any term to the preceding approaches x as its limit
in each case. The first series diverges for aj = ±1. The second
series diverges for x = 1, and converges for x = — 1. The third con
verges absolutely for x = ± 1.
Note. The statement of Abel's theorem may be made more general,
for it is sufficient for the argument that the absolute value of any
term of the series
«0 + alX0 + ---- 1- anX0 H ----
be less than a fixed number. Whenever this condition is satisfied,
the series (2) converges absolutely for any value of x whose absolute
value is less than \x0 .
The number R is connected in a very simple way with the number to defined
in § 160, which is the greatest limit of the Sequence
For if we consider the analogous sequence
378 SPECIAL SERIES [IX, § 178
it is evident that the greatest limit of the terms of the new sequence is <aX. The
sequence (1) therefore converges if X < 1/w, and diverges if X > 1/w ; hence
178. Continuity of a power series. Let f(x) be the sum of a power
series which converges in the interval from — R to + R,
(3) f(x) = a0
and let R' be a positive number less than R. We shall first show
that the series (3) converges uniformly in the interval from — R'
to + R'. For, if the absolute value of x is less than R', the
remainder Rn
Rn = an+lxn + l + ••• + an+pxn + P + •••
of the series (3) is less in absolute value than the remainder
4.+I***1 +-*.+;***•*•••
of the corresponding series (1). But the series (1) converges for
X = R', since R' < R. Consequently a number N may be found
such that the latter remainder will be less than any preassigned
positive number c whenever n ^ N. Hence Rn \ < c whenever n > N
provided that \x\ < R'.
It follows that the sum f(x} of the given series is a continuous
function of x for all values of x between — R and + R. For, let x0
be any number whose absolute value is less than R. It is evident
that a number R1 may be found which is less than R and greater
than \x0\. Then the series converges uniformly in the interval
(_ R', + R'~), as we have just seen, and hence the sum f(x) of the
series is continuous for the value x0, since x0 belongs to the interval
in question.
This proof does not apply to the end points + R and — R of the
interval of convergence. The function f(x} remains continuous,
however, provided that the series converges for those values.
Indeed, Abel showed that if the series (3) converges for x = R, its
sum for x = R is the limit which the sum /(#) of the series approaches
as x approaches R through values less than J?.f
Let S be the sum of the convergent series
S = «0 + al R + a2R* -\ ---- + anR" H ---- ,
* This theorem was proved by Cauchy in his Cours d' Analyse. It was rediscovered
by Hadamard in his thesis.
t As stated above, these theorems can be immediately generalized to the case of
series of imaginary terms. In this case, however, care is necessary in formulating
the generalization. See Vol. II, § 266. — TRANS.
IX, §m] POWER SERIES 379
and let n be a positive integer such that any one of the sums
is less than a preassigned positive number e. If we set x = R0, and
then let 0 increase from 0 to 1, a; will increase from 0 to R, and we
shall have
/(z) =/(0J?) = a0 + al0R + a202£2 + • • • + an$»R" 4. ....
If n be chosen as above, we may write
•S -f(x) = aiR(l - 0) + «2#2(1 - 02) + • • • + anRn(l - 0")
(4)- +an+lRn + l + --- + an+pRn+? + •-.
and the absolute value of the sum of the series in the second line can
not exceed e. On the other hand, the numbers On + l, 0B + 2, • •-, 0H+P
form a decreasing sequence. Hence, by Abel's lemma proved in § 75,
we shall have
\an + l6n + lRn + l -i ---- + an+pOn+pRn+p < 6n + lt < e.
It follows that the absolute value of the sum of the series in the
third line cannot exceed e. Finally, the first line of the right-hand
side of the equation (4) is a polynomial of degree n in 0 which
vanishes when 0=1. Therefore another positive number rj may be
found such that the absolute value of this polynomial is less than c
whenever 0 lies between 1 — r\ and unity. Hence for all such values
of 0 we shall have
\S~f(x)\<3e.
But e is an arbitrarily preassigned positive number. Hence f(x)
approaches S as its limit as x approaches R.
In a similar manner it may be shown that if the series (3) con
verges for x = — R, the sum of the series for x = — R is equal to
the limit which /(x) approaches as x approaches — R through values
greater than — R. Indeed, if we replace x by — x, this case reduces
to the preceding.
An application. This theorem enables us to complete the results of § 168
regarding the multiplication of series. Let
(5) S =UQ + UI + UZ-\ ---- '+ un H ---- ,
(6) S' = v0 + vl + t>2 H ----- h v« + • • •
be two convergent series, neither of which converges absolutely. The series
(7) u0»o + («ot>i 4- Uiu0) H ---- + (uo«» H ----- 1- «n»o) H ----
380 SPECIAL SERIES [IX, §179
may converge or diverge. If it converges, its sum S is equal to the product of
the sums of the two given series, i.e. S = <S<S'. For, let us consider the three
power series
f(x) =
tf>(x) = v0 + vix + • • • + vnxn H ---- ,
-\ ----- h
Each of these series converges, by hypothesis, when z = 1. Hence each of them
converges absolutely for any value of x between — 1 and + 1. For any such
value of x Cauchy's theorem regarding the multiplication of series applies and
gives us the equation
(8) /(*)0(z) = *(z).
By Abel's theorem, as x approaches unity the three functions /(x), #(x), \ft(x)
approach S, S', and 2, respectively. Since the two sides of the equation (8)
meanwhile remain equal, we shall have, in the limit, S = SS'.
The theorem remains true for series whose terms are imaginary, and the proof
follows precisely the same lines.
179. Successive derivatives of a power series. If a power series
f(x) = a0 + avx + a2x2 -\ ----- h «„*" H ----
which converges in the interval (— R, -f R) be differentiated term
by term, the resulting power series
converges in the same interval. In order to prove this, it will be
sufficient to show that the series of absolute values of the terms of
the new series,
Av + 2A9X-\ h nAnXn~l -\ ,
where At = |a,.| and X = \x\, converges for X<R and diverges for
X>R.
For the first part let us suppose that X < R, and let R' be a num
ber between X and R, X < R' < R. Then the auxiliary series
— h — — 4- — (— Y 4- 4- — I—}*'1 4
converges, for the ratio of any term to the preceding approaches
X/R', which is less than unity. Multiplying the successive terms
of this series, respectively, by the factors
IX, §179] POWER SERIES 381
each of which is less than a certain fixed number, since R' <R, we
obtain a new series
which also evidently converges.
The proof of the second part is similar to the above. If the series
A1 + 2A2Xl + ••• + nAnXr' + •••>
where Xt is greater than R, were convergent, the series
AlXl + 2AtX* + .-- + nAnXnl +•••
+ »
would converge also, and consequently the series 2^nX" would con
verge, since each of its terms is less than the corresponding term of
the preceding series. Then R would not be the upper limit of the
values of X for which the series (1) converges.
The sum /j (a:) of the series (9) is therefore a continuous function
of the variable x inside the same interval. Since this series con
verges uniformly in any interval (— R', -f- R'), where R' < R)f1(x)
is the derivative of f(x) throughout such an interval, by § 174.
Since R' may be chosen as near R as we please, we may assert that
the function f(x) possesses a derivative for any value of x between
— R and -f R, and that that derivative is represented by the series
obtained by differentiating the given series term by term : *
(10) /(*) = «! + 2a2x + ••• + nanx*-1 + ....
Repeating the above reasoning for the series (10), we see that f(x)
has a second derivative,
/"(a) = 2a2 + 6a3x + . . . + w(n - 1) anxn~2 + • • -,
and so forth. The function f(x) possesses an unlimited sequence of
derivatives for any value of x inside the interval (—/£, + R), and
these derivatives are represented by the series obtained by differen
tiating the given series successively term by term :
(11) f<*\x) = 1.2---nan + 2.3---n(n + l~)an + lx-i ---- .
If we set x = 0 in these formulae, we find
or, in general,
* Although the corresponding theorem is true for series of imaginary terms, the
proof follows somewhat different lines. See Vol. II, § 266. — TRANS.
382 SPECIAL SERIES [IX, § 179
The development of /(#) thus obtained is identical with the develop
ment given by Maclaurin's formula :
/(*) =/(0) + f /'(<>) + f^/"(<>) + • • • + iTf^/^O) + • • -.
The coefficients a0) al9 •••, an, ••• are equal, except for certain
numerical factors, to the values of the function f(x) and its succes
sive derivatives for x = 0. It follows that no function can have two
distinct developments in power series.
Similarly, if a power series be integrated term by term, a new
power series is obtained which has an arbitrary constant term and
which converges in the same interval as the given series, the given
series being the derivative of the new series. If we integrate again,
we obtain a third series whose first two terms are arbitrary ; and so
forth.
Examples. 1) The geometrical progression
1 — X + X* — Xs -\ ----- \- (—l)nXn -\ ---- ,
whose ratio is — x, converges for every value of x between — 1 and
+ 1, and its sum is 1/(1 + x). Integrating it term by term between
the limits 0 and x, where x < 1, we obtain again the development
of log (1 + x*) found in § 49 :
This formula holds also for x = 1, for the series on the right con
verges when x = 1.
2) For any value of x between — 1 and + 1 we may write
^-^ = 1 - x 2 + *4 - x 6 + ... + (- 1 )" x2 " + ••••
Integrating this series term by term between the limits 0 and x,
where Ice I < 1, we find
Since the new series converges for x = 1, it follows that
TT 111 1
IX, §17'.']
row EH SERIES
383
3) Let F(x) be the sum of the convergent series
m m(m — 1)
+
i(m —!)••• (in,—
where m is any number whatever and | a: < 1. Then we shall have
[m— 1
! + — * +
(m -1). ••(m — p +1)
< 1.2.. -1 J
Let us multiply each side by (1+ a;) and then collect the terms in
like powers of x. Using the identity
(m — 1) • • • (m — p +1) (m— !)•• -(m— />) _ m(?rc —1) • • • (m — p -f-1)
1.2.--OJ-1) 1.2-.-.P 1.2--.J9
which is easily verified, we find the formula
,
P ,
or
From this result we find, successively,
F'(x) _ m
~F(x)~l+x
log [/•'(»)] = m log (1 + a-) + log
or
To determine the constant C we need merely notice that F(0) = l.
Hence C = 1. This gives the development of (1 + a*)'" found in § 50 :
Y»_1.™ . m(m-l)...(m-.p+l) ,
4) Replacing a; by — x2 and /«, by —1/2 in the last formula above,
we find
,1
=1+ -
1.3
-— -
2.4
,
H ----- 1
1.3. 5- ..(27t-
v — ^
2 . 4 . 6 • • • 2n
This formula holds for any value of x between —1 and +1. Inte
grating both sides between the limits 0 and x, where | x < 1, we
obtain the following development for the arcsine :
x , 1 a-3 1.3a;5 , , 1 . 3 . 5 • - - (2n -1) xz" + l
arc sm x = - -f 7: -^ + ^ T -=- H ----- — — —
0 . g
2.4.6---2n
384
SPECIAL SERIES
[IX, § 180
180. Extension of Taylor's series. Let/(x) be the sum of a power
series which converges in the interval (— R, + R~), a-0 a point inside
that interval, and x0 + h another point of the same interval such
that |a;0| + | h\ < R. The series whose sum is f(x0 + A),
«o + ai(«0 + A) + a2(x0 + A)2 H \- an(x0 + A)" -] ,
may be replaced by the double series obtained by developing each
of the powers of (x0 + A) and writing the terms in the same power
of h upon the same line :
(12)
-f- 2a2x0h
n
n(n — 1)
1.2
This double series converges absolutely. For if each of its terms
be replaced by its absolute value, a new double series of positive
terms is obtained :
(13)
If we add the elements in any one column, we obtain a series
(-
^n\X0
n i _
••• +
n An x0
n~l\h\ +••
1
i(n — 1)
n-2| ^|2_1_. .
• •• T
1.2 ff°
which converges, since we have supposed that | x0 \ + 1 h \ < R. Hence
the array (12) may be summed by rows or by columns. Taking
the sums of the columns, we obtain f(x0 + A). Taking the sums
of the rows, the resulting series is arranged according to powers of
A, and the coefficients of A, h2, ••• are f'(x0),f"(x0)/2l, •••, respec
tively. Hence we may write
(14)
A) =/(.r0) +
.L .
71
if we assume that | A [ < R — | x0 \ .
This formula surely holds inside the interval from x0 — R +\x0\
to x0 + R — \x0 , but it may happen that the series on the right
converges in a larger interval. As an example consider the function
IX, §180] POWER SERIES 386
(1 + x~)m, where m is not a positive integer. The development
according to powers of x holds for all values of x between — 1 and
+ 1. Let x0 be a value of x which lies in that interval. Then we
may write
(i + x}m = (i + x0 + x - X«Y = (i + <r (i + *)m,
where
z =
X — X
l+X0
We may now develop (1 + z)m according to powers of 2, and this
new development will hold whenever \z\ < 1, i.e. for all values of x
between —1 and 1 + 2x0. If x0 is positive, the new interval will be
larger than the former interval (—1, +!)• Hence the new formula
enables us to calculate the values of the function for values of the
variable which lie outside the original interval. Further investiga
tion of this remark leads to an extremely important notion, — that
of analytic extension. We shall consider this subject in the second
volume.
Note. It is evident that the theorems proved for series arranged
according to positive powers of a variable x may be extended immedi
ately to series arranged according to positive powers of x — a, or,
more generally still, to series arranged according to positive powers
of any continuous function <£(x) whatever. We need only consider
them as composite functions, <f>(x) being the auxiliary function.
Thus a series arranged according to positive powers of I/a; con
verges for all values of x which exceed a certain positive constant in
absolute value, and it represents a continuous function of x for all
such values of the variable. The function Va/2 — a, for example, may
be written in the form ± x(\ — a/a:2)*. The expression (1 — a/ar2)*
may be developed according to powers of I/a;2 for all values of x
which exceed V a in absolute value. This gives the formula
1 . 2 . 3 • • - (2P - 3) a"
2.4.6---2p a-2'-1'
which constitutes a valid development of Va;2 — a whenever x > Va.
When x < — Va, the same series converges and represents the func
tion — Va:2 — a. This formula may be used advantageously to obtain
a development for the square root of an integer whenever the first
perfect square which exceeds that integer is known.
386 SPECIAL SERIES [IX, § 181
181. Dominant functions. The theorems proved above establish a
close analogy between polynomials and power series. Let (— r, + r)
be the least of the intervals of convergence of several given power
series /i (x), /2 (x), ••-,/» (#)• When |cc|<r, each of these series
converges absolutely, and they may be added or multiplied together
by the ordinary rules for polynomials. In general, any integral poly
nomial in /1(x),/2(a;), •••,/„(«) may be developed in a convergent
power series in the same interval.
For purposes of generalization we shall now define certain expres
sions which will be useful in what follows. Let f(x~) be a power
series
f(x) = a0 + aj_x + a2x2 -) h anx" H ,
and let <f>(x) be another power series with positive coefficients
<f,(x) = a0 + a^x + a2x2 H f- anxn -\
which converges in a suitable interval. Then the function <£(a;) is
said to dominate * the function f(x) if each of the coefficients an is
greater than the absolute value of the corresponding coefficient of
/(*):
\<10\<C*0,
Poincare has proposed the notation
f(x) < <£(*)
to express the relation which exists between the two functions f(x)
and <f>(x~).
The utility of these dominant functions is based upon the fol
lowing fact, which is an immediate consequence of the definition.
Let P(a0, a\, •••, «„) be a polynomial in the first n -f- 1 coefficients
of f(x) whose coefficients are all real and positive. If the quanti
ties aQ, «!, ••-, an be replaced by the corresponding coefficients of
<£(#), it is clear that we shall have
|P(Oo, a1} ••-, an~)\<P(a0, a1} •••, an).
For instance, if the function <f>(x) dominates the function /(#),
the series which represents [<£(:r)]2 will dominate [/(ic)]2, and so
on. In general, [<£(#)]" wiU dominate [/(«)]". Similarly, if <f> and
^>! are dominant functions for / and /t , respectively, the product <£<£j
will dominate the product jff\ ; and so forth.
*This expression will be used as a translation of the phrase " <p(x) est majorante
pour la fonction /(a;)." Likewise, "dominant functions " will be used for " fonctions
majorantes." — TRANS.
IX, § 181]
POWER SERIES 387
Given a power series/(x) which converges in an interval (— R, + R),
the problem of determining a dominant function is of course indeter
minate. But it is convenient in what follows to make the domi
nant function as simple as possible. Let r be any number less than
R and arbitrarily near R. Since the given series converges for x = r,
the absolute value of its terms will have an upper limit, which we
shall call M. Then we may write, for any value of n,
or
Hence the series
x Mxn M
M + M- '
r r" x
r
whose general term is M(xn/rn), dominates the given function /(x).
This is the dominant function most frequently used. If the series
/(x) contains no constant term, the function
M
may be taken as a dominant function.
It is evident that r may be assigned any value less than R, and
that M decreases, in general, with r. But M can never be less than
A0. If A0 is not zero, a number p less than R can always be found
such that the function -40/(1 — x/p) dominates the function /(x).
For, let the series
M + M - + M ^ + • • • + M ^ + • • • »
r r2 r"
where M > A0, be a first dominant function. If p be a number less
than rA0/M and n > 1, we shall have
whence |anpn| < A0. On the other hand, |ac| = A0. Hence the series
dominates the function f(x). We shall make use of this fact pres
ently. More generally still, any number whatever which is greater
than or equal to A0 may be used in place of M.
388 SPECIAL SERIES [IX, § 182
It may be shown in a similar manner that if a0 = 0, the function
is a dominant function, where /t is any positive number whatever.
Note. The knowledge of a geometrical progression which dominates the func
tion f(x) also enables us to estimate the error made in replacing the function
f(x) by the sum of the first n + 1 terms of the series. If the series M/(l — x/r)
dominates /(z), it is evident that the remainder
of the given series is less in absolute value than the corresponding remainder
of the dominant series. It follows that the error in question will be less than
(T1
jfW —
182. Substitution of one series in another. Let
(15) * =/(>/) = «o + «iy 4- • • • + any + •••
be a series arranged according to powers of a variable y which con
verges whenever \y\<. R. Again let
/ 1 A \ — -JL ^ \ TitA I t A w. i
be another series, which converges in the interval (— r, + r). If
y> y2) y8) ••• in the series (15) be replaced by their developments in
series arranged according to powers of x from (16), a double series
+
I ,. 7, ., l O« 7, 7, ~.
(17)
a2 (b\
is obtained. We shall now investigate the conditions under which
this double series converges absolutely. In the first place, it is
necessary that the series written in the first row,
IX, §182] POWER SERIES 389
should converge absolutely, i.e. that \b0\ should be less than R.* This
condition is also sufficient. For if it is satisfied, the function <(>(x)
will be dominated by an expression of the form m/(l — x/p), where
m is any positive number greater than \b0\ and where p < r. We
may therefore suppose that m is less than R. Let R' be another
positive number which lies between m and R. Then the function
f(y) is dominated by an expression of the form
y R' R'2
R'
If y be replaced by wi/(l — x/p) in this last series, and the powers
of y be developed according to increasing powers of x by the binomial
theorem, a new double series
(18)
M ^
m x
M j^ + '-' + nMt-
is obtained, each of whose coefficients is positive and greater than
the absolute value of the corresponding coefficients in the array (17),
since each of the coefficients in (17) is formed from the coefficients
a0, a*!, a2, • • •, b0) bi, b2, • • -by means of additions and multiplications
only. The double series (17) therefore converges absolutely pro
vided the double series (18) converges absolutely. If x be replaced
by its absolute value in the series (18), a necessary condition for abso
lute convergence is that each of the series formed of the terms in any
one column should converge, i.e. that \x\ < p. If this condition be
satisfied, the sum of the terms in the (n + l)th column is equal to
Then a further necessary condition is that we should have
or
(19)
* The case in which the series (15) converges for y = R (see § 177) will be neglected
in what follows. — TRANS.
390 SPECIAL SERIES [IX, § 182
Since this latter condition includes the former, \x\ < p, it follows
that it is a necessary and sufficient condition for the absolute con
vergence of the double series (18). The double series (17) will
therefore converge absolutely for values of x which satisfy the
inequality (19). It is to be noticed that the series <j>(x) converges
for all these values of x, and that the corresponding value of y is
less than R' in absolute value. For the inequalities
i.t x, . m. \x\ m
1 ( ! '
necessitate the inequality $(x)\<.R'. Taking the sum of the series
(17) by columns, we find
that is, f[$(x)~\- On the other hand, adding by rows, we obtain a
series arranged according to powers of x. Hence we may write
where the coefficients c0, ci} c2, • • • are given by the formulae
CQ = <ZQ -f- fti w0 + ft2 t>o ~r • • • ~r <^n "o i ' ' ' >
(21)
which are easily verified.
The formula (20) has been established only for values of x which
satisfy the inequality (19), but the latter merely gives an under
limit of the size of the interval in which the formula holds. It may
be valid in a much larger interval. This raises a question whose
solution requires a knowledge of functions of a complex variable.
We shall return to it later.
Special cases. 1) Since the number R' which occurs in (19) may
be taken as near R as we please, the formula (20) holds whenever x
satisfies the inequality \x < p(l— m/R). Hence, if the series (15)
converges for any value of y whatever, R may be thought of as infinite,
p may be taken as near r as we please, and the formula (20) applies
whenever |a;| < r, that is, in the same interval in which the series
(16) converges. In particular, if the series (16) converges for all
values of x, and (15) converges for all values of y, the formula (20)
is valid for all values of x.
IX, § 182]
POWER SERIES 391
2) When the constant term b0 of the series (16) is zero, the func
tion <£(z) is dominated by an expression of the form
l-x-
p
where p < r and where m is any positive number whatever. An
argument similar to that used in the general case shows that the
formula (20) holds in this case whenever x satisfies the inequality
(22) z|<P— ,
v ' r R' + m
where R' is as near to R as we please. The corresponding interval
of validity is larger than that given by the inequality (19).
This special case often arises in practice. The inequality
\b0\ < R is evidently satisfied, and the coefficients cn depend upon
Examples. 1) Cauchy gave a method for obtaining the binomial theorem from
the development of log(l + x). Setting
y =
we may write
whence, substituting the first expansion in the second,
If the right-hand side be arranged according to powers of x, it is evident that
the coefficient of x" will be a polynomial of degree n in ju, which we shall call
Pn(n)- This polynomial must vanish when ^ = 0, 1, 2, • • •, n — 1, and must
reduce to unity when ^ = n. These facts completely determine Pn in the form
.M p _M(M-l)---(M-n+l)
P" - ~"""
2) Setting z = (1 + z)1/*, where x lies between - 1 and + 1, we may write
where
392 SPECIAL SERIES [IX, §183
The first expansion is valid for all values of y, and the second is valid whenever
\x\< 1. Hence the formula obtained by substituting the second expansion in
the first holds for any value of x between — 1 and + 1. The first two terms of
this formula are
T__ .
It follows that (1 + x)1/* approaches e through values less than e as z approaches
zero through positive values.
183. Division of power series. Let us first consider the reciprocal
f(x\ — _ _ _
A } l+blX + b2x* + ...
of a power series which begins with unity and which converges in
the interval (— r, -f- ?•). Setting
y = bvx + b2x2 H ---- ,
we may write
/(*) = r+~y =1~ y + y2 - y* + •••'
whence, substituting the first development in the second, we obtain
an expansion for f(x) in power series,
(25) f(x) = 1 - blX + (b\ - 6.) *2 + • • -,
which holds inside a certain interval. In a similar manner a devel
opment may be obtained for the reciprocal of any power series
whose constant term is different from zero.
Let us now try to develop the quotient of two convergent power
series
$(x) bti + biX + b3x* -\ ----
If b0 is not zero, this quotient may be written in the form
= (a +a x + a 2x* + -- •) X
Then by the case just treated the left-hand side of this equation is the
product of two convergent power series. Hence it may be written
in the form of a power series which converges near the origin :
-f
Clearing of fractions and equating the coefficients of like powers
of x, we find the formulae
IX, §184] POWER SERIES 393
(27) an = l0cH + biCH_l+--- + bHc9, n = 0, 1, 2, ••-,
from which the coefficients c0, clt •••, cn may be calculated succes
sively. It will be noticed that these coefficients are the same as
those we should obtain by performing the division indicated by the
ordinary rule for the division of polynomials arranged according to
increasing powers of x. .
If b0 = 0, the result is different. Let us suppose for generality
that ij/(x~) = xk fa (x), where A; is a positive integer and \l/\(x) is a
power series whose constant term is not zero. Then we may write
and by the above we shall have also
It follows that the given quotient is expressible in the form
\ / "77 \ = * *t—I < ' ' ' * ,~ ' ck -j- Cj. + j a; -p • • • ,
where the right-hand side is the sum of a rational fraction which
becomes infinite for x = 0 and a power series which converges near
the origin.
Note. In order to calculate the successive powers of a power series, it is con
venient to proceed as follows. Assuming the identity
(a0 + a\x + • • • + anxn + • • •)'" = c0 + ctx + • • • + cnxn + • • •,
let us take the logarithmic derivative of each side and then clear of fractions.
This leads to the new identity
• • • + nanxn-1 + • • -)(c0 + Cix + • • • 4- cnxn + • • •)
(29)
a,(x"
The coefficients of the various powers of x are easily calculated. Equat
ing coefficients of like powers, we find a sequence of formulae from which
CQ, Ci, • • •, cn, • • • may be found successively if c0 be known. It is evident that
184. Development of 1/Vl — 2xz -|- z2. Let us develop 1/Vl - 2xz + z2
according to powers of z. Setting y = 2xz — z2, we shall have, when \y\< 1,
or
(30) 1 = 1 + 2xz ~ 28 + - (2xz - z2)2 +
Vl^2xzTP 2 8
394 SPECIAL SERIES [ix,§i86
Collecting the terms which are divisible by the same power of z, we obtain an
expansion of the form
(31) l -- = p0 + plZ + pzZ2 + ... + pnZn + ...,
VI - ?xz + z'2
where
2 *
and where, in general, Pn is a polynomial of the nth degree in x. These poly
nomials may be determined successively by means of a recurrent formula. Dif
ferentiating the equation (31) with respect to z, we find
(1 - 2xz + z2)*
or, by the equation (31),
(x - z)(P0 + PIZ + h Pn*n + • • •) = (1 - 2xz + z2)(Pi + 2P2z H ) .
Equating the coefficients of z", we obtain the desired recurrent formula
This equation is identical with the relation between three consecutive Legendre
polynomials (§ 88), and moreover P0 = X0 , P! = X\ , P2 = X2 . Hence Pn = Xn
for all values of n, and the formula (31) may be written
Vl - 2xz + z2 ~
where Xn is the Legendre polynomial of the nth order
-V id i\_n
-&-n = \\&* — 1)1 •
2 . 4 . 6 • • • 2n dxn
We shall find later the interval in which this formula holds.
II. POWER SERIES IN SEVERAL VARIABLES
185. General principles. The properties of power series of a single
variable may be extended easily to power series in several independ
ent variables. Let us first consider a double series 2tamnxmyn, where
the integers m and n vary from zero to + oo and where the coeffi
cients amn may have either sign. If no element of this series exceeds
a certain positive constant in absolute value for a set of values
x — xo> y = 2/0) the series converges absolutely for all values of x and
y which satisfy the inequalities \x\ < \x0\, \y\ < \y0\.
For, suppose that the inequality
M
I amn x™Vo \<M or | amu I < , ,m. ,„
I xo \ \yo\
IX, §185] DOUBLE POWER SERIES 395
is satisfied for all sets of values of m and n. Then the absolute value
of the general element of the double series 2amn.rmy is less than the
corresponding element of the double series 2M\x/x0\m\y/y0\n. But
the latter series converges whenever |#|<|a:0|, |y|<|y0|> and its
sum is
M
2/o
as we see by taking the sums of the elements by columns and then
adding these sums.
Let r and p be two positive numbers for which the double series
2 amn\rmpn converges, and let R denote the rectangle formed by the
four straight lines x = r, x = — r, y = p, y — — p. For every point
inside this rectangle or upon one of its sides no element of the
double series
exceeds the corresponding element of the series ^\amn\rmpn in abso
lute value. Hence the series (33) converges absolutely and uni
formly inside of R, and it therefore defines a continuous function
of the two variables x and y inside that region.
It may be shown, as for series in a single variable, that the
double series obtained by any number of term-by-term differen
tiations converges absolutely and uniformly inside the rectangle
bounded by the lines x = r — c, x = — r + c, y = p — e', y = — p + e',
where c and e' are any positive numbers less than r and p, respec
tively. These series represent the various partial derivatives of
F(x,y). For example, the sum of the series '2tmamnxm-lyn is equal
to cF/dx. For if the elements of the two series be arranged accord
ing to increasing powers of x, each element of the second series is
equal to the derivative of the corresponding element of the first.
Likewise, the partial derivative dm+nF/dxmdyn is equal to the sum
of a double series whose constant factor is aranl . 2 • • • m . 1 . 2 • • • n.
Hence the coefficients amn are equal to the values of the correspond
ing derivatives of the function F(x, y) at the point x = y — 0, except
for certain numerical factors, and the formula (33) may be written
in the form
F(x, y)=
\ >yj
/8m + "F\
396 SPECIAL SERIES [IX, § 186
It follows, incidentally, that no function of two variables can have
two distinct developments in power series.
If the elements of the double series be collected according to
their degrees in x and y, a simple series is obtained :
(35) F(x, y) = <£0 + fa + <f>2 + • • • + <£„ + • • • ,
where </>„ is a homogeneous polynomial of the nth degree in x and
y which may be written, symbolically,
The preceding development therefore coincides with that given by
Taylor's series (§ 51).
Let (a-0, y0~) be a point inside the rectangle R, and (x0 + h, y0 + k)
be a neighboring point such that | x0 \ + 1 h \ < r, \y0\ + \k\ < p. Then
for any point inside the rectangle formed by the lines
* = ar0±[r-|a:0|], y = y» ± [p - |y0|],
the function F(x, y) may be developed in a power series arranged
according to positive powers of x — xv and y — y0 :
»F\
ZdTmf)i/n/x=x°
1.2 ,! 1. ';*..»*"*••
For if each element of the double series
be replaced by its development in powers of h and k, the new multi
ple series will converge absolutely under the hypotheses. Arrang
ing the elements of this new series according to powers of h and k,
we obtain the formula (36).
The reader will be able to show without difficulty that all the
preceding arguments and theorems hold without essential altera
tion for power series in any number of variables whatever.
186. Dominant functions. Given a power series f(x, y, z, •• •) in n
variables, we shall say that another series in n variables </>(«, y, z, • • •)
dominates the first series if each coefficient of <j>(x, y,z, • • •) is positive
and greater than the absolute value of the corresponding coefficient
of f(x, y, z, • • •). The argument in § 185 depends essentially upon
IX, §186] DOUBLE POWER SERIES 397
the use of a dominant function. For if the series 2,\amnx'"y"\ con
verges for x = r, y = p, the function
M /x\m /vY*
d>(x, y} = = 3/2 ( - ) ( - >
/ x\ I y\ \r/ \P/
where M is greater than any coefficient in the series 2,\amnrmp"\,
dominates the series 2,amnxmyn. The function
is another dominant function. For the coefficient of xmyn in 1/^(3*, y)
is equal to the coefficient of the corresponding term in the expan
sion of M(x/r -+- y/p}m+n, and therefore it is at least equal to the
coefficient of xmyn in <f>(x, y).
Similarly, a triple series
which converges absolutely for x = r, y = r', z = r", where r, r', r"
are three positive numbers, is dominated by an expression of the
form
M
y>
'l_±Ul_£\/l_±-
and also by any one of the expressions
M M
x\[~. /?/ z \ ~1
- ) !-(-, + — )
r/\_ \r r"/ J
If f(x, y, z) contains no constant term, any one of the preceding expres
sions diminished by M may be selected as a dominant function.
The theorem regarding the substitution of one power series in
another (§ 182) may be extended to power series in several variables.
If each of the variables in a convergent power series in p variables
y\i y-ii ' ' ') yp b& replaced by a convergent power series in q variables
x\j xii •*•> xq which has no constant term, the result of the substitu
tion may be written in the form of a power series arranged according
to powers of xl, x2, • • • , xq, provided that the absolute value of each
of these variables is less than a certain constant.
398 SPECIAL SERIES [IX, § 186
Since the proof of the theorem is essentially the same for any
number of variables, we shall restrict ourselves for definiteness to
the following particular case. Let
(37)
be a power series which converges whenever y < r and z < r', and let
(38) (y=-t>i*--b2x^ -bm* ..,
f £ — Cj 00 ~~r~ ^ 2 **^ ~T~ " * * j* t« »<C ~J~ • • •
be two series without constant terms both of which converge if the
absolute value of x does not exceed p. If y and z in the series (37)
be replaced by their developments from (38), the term in ymz" becomes
a new power series in x, and the double series (37) becomes a triple
series, each of whose coefficients may be calculated from the coeffi
cients amn, bn, and cn by means of additions and multiplications
only. It remains to be shown that this triple series converges abso
lutely when the absolute value of x does not exceed a certain con
stant, from which it would then follow that the series could be
arranged according to increasing powers of x. In the first place,
the function f(y, z) is dominated by the function
(39)
and both of the series (38) are dominated by an expression of the form
N ^T^ (x\n
(40) N= > .V (-) ,
, x L*i \P/
1 71= 1
P
where M and N are two positive numbers. If y and z in the double
series (39) be replaced by the function (40) and each of the products
ymzn be developed in powers of x, each of the coefficients of the result
ing triple series will be positive and greater than the absolute value
of the corresponding coefficient in the triple series found above. It
will therefore be sufficient to show that this new triple series con
verges for sufficiently small positive values of x. Now the sum of
the terms which arise from the expansion of any term ymz* of the
series (39) is
M-«
AT'-" Vp/
IX, §187] REAL ANALYTIC FUNCTIONS 399
which is the general term of the series obtained by multiplying the
two series
x \m i x
P
y Mry _p_
L*\rl 1_x_
term by term, except for the constant factor M. Both of the latter
series converge if x satisfies both of the inequalities
x < - ' x <
It follows that all the series considered will converge absolutely,
and therefore that the original triple series may be arranged accord
ing to positive powers of x, whenever the absolute value of x is less
than the smaller of the two numbers pr/(r + JV) and pr'/(r' + N).
Note. The theorem remains valid when the series (38) contain
constant terms b0 and c0, provided that \b0\< r and c0 < r'. For
the expansion (37) may be replaced by a series arranged according
to powers of y — b0 and z — c0, by § 185, which reduces the discus
sion to the case just treated.
III. IMPLICIT FUNCTIONS
ANALYTIC CURVES AND SURFACES
187. Implicit functions of a single variable. The existence of implicit
functions has already been established (Chapter II, § 20 et ff.) under
certain conditions regarding continuity. When the left-hand sides
of the given equations are power series, more thorough investigation
is possible, as we shall proceed to show.
Let F(x, y~) = 0 be an equation whose left-hand side can be developed
in a convergent power series arranged according to increasing powers
of x — x0 and y — yQ, where the constant term is zero and the coeffi
cient of y — y0 is different from zero. Then the equation has one and
only one root which approaches >/n as x approaches XQ, and that root
can be developed in a power series arranged according to powers of
x - x0.
For simplicity let us suppose that x0 = y0 = 0, which amounts to
moving the origin of coordinates. Transposing the term of the first
degree in y, we may write the given equation in the form
(41) y = f(x, y} = a^x + a20a;2 + anxy + a^y* H ,
400 SPECIAL SERIES [IX, § 187
where the terms not written down are of degrees greater than the
second. We shall first show that this equation can be formally sat
isfied by replacing y by a series of the form
(42) y = Clx + c2x* + -.- + cnxn + ---
if the rules for operation on convergent series be applied to the series
on the right. For, making the substitution and comparing the coeffi
cients of x, we find the equations
ci = aio> c2 = aso-f anCj + a02cf, •••;
and, in general, cn can be expressed in terms of the preceding c's
and the coefficients aik, where i + k < n, by means of additions and
multiplications only. Thus we may write
(43) cn = Pn(a10, aw> an, • ••, a0n),
where Pn is a polynomial each of whose coefficients is a positive
integer. The validity of the operations performed will be estab
lished if we can show that the series (42) determined in this way
converges for all sufficiently small values of x. We shall do this by
means of a device which is frequently used. Its conception is due
to Cauchy, and it is based essentially upon the idea of dominant
functions. Let
be a function which dominates the f unction /(x, y), where #00 = bol = 0
and where bmn is positive and at least equal to amn \ . Let us then
consider the auxiliary equation
(41') Y = 4>(x, Y) = 2bmnxmYn
and try to find a solution of this equation of the form
(42') Y=C,x+ C2x* + ... + cnz" + --..
The values of the coefficients C15 C2, • • • can be determined as above,
and are
Cl = blo, C2 = b20 + bnCl -{-b02Cl, ••-,
and in general
(43') C. = Pn(bw,b20, •••,*„„)•
It is evident from a comparison of the formulae (43) and (43')
that \cn\ < Cn, since each of the coefficients of the polynomial Pn is
positive and \amn\^bmn. Hence the series (42) surely converges
EX, §187] REAL ANALYTIC FUNCTIONS 401
whenever the series (42') converges. Now we may select for the
dominant function <j>(x, Y) the function
Y
" M ~ M 7'
where M, r, and p are three positive numbers. Then the auxiliary
equation (41') becomes, after clearing of fractions,
p + M p + M -L _ *
r
This equation has a root which vanishes for x = 0, namely :
F =
The quantity under the radical may be written in the form
where
P
Hence the root Y may be written
']
It follows that this root Y may be developed in a series which con
verges in the interval (— a, -f a), and this development must coin
cide with that which we should obtain by direct substitution, that
is, with (42'). Accordingly the series (42) converges, a fortiori, in
the interval (— a, + a). This is, however, merely a lower limit of
the true interval of convergence of the series (42), which may be
very much larger.
It is evident from the manner in which the coefficients cn were
determined that the sum of the series (42) satisfies the equation (41).
Let us write the equation F(x, y) in the form y — f(x, y) = 0, and
let y = P(x) be the root just found. Then if P(x) + « be substi
tuted for y in F(x, y), and the result be arranged according to
powers of x and z, each term must be divisible by z, since the whole
expression vanishes when z = 0 for any value of x. We shall have
then F[x, P(x) -f *] = sQ(#, «) , where Q(x, z) is a power series in x
402 SPECIAL SERIES [IX, § 188
and z. Finally, if z be replaced by y — P(x} in Q(x, z), we obtain
the identity
where the constant term of Qt must be unity, since the coefficient
of y on the left-hand side is unity. Hence we may write
(44) F(x, y) = [y - P(x)] (1 + ax + fa + ...).
This decomposition of F(x, y) into a product of two factors is due
to Weierstrass. It exhibits the root y = P(x), and also shows that
there is no other root of the equation F(x, y) = 0 which vanishes
with x, since the second factor does not approach zero with x and y.
Note. The preceding method for determining the coefficients cn is
essentially the same as that given in § 46. But it is now evident
that the series obtained by carrying on the process indefinitely is
convergent.
188. The general theorem. Let us now consider a system of p equa
tions in p -f q variables.
i(«i,*a» •••»*«'; yi,yaj •••JyP) = 0,
2(x1}x2, '-',x9i yi,y2, •-,y,)= 0,
' ........... '
(*u *•* •••>*«; yi>y2, •••»%)== o,
where each of the functions F1} F2, • • •, Fp vanishes when xi — yk = 0,
and is developable in power series near that point. We shall further
suppose that the Jacobian D(Fl} F2, ••-, Fp)/D(y1, y2, •••, yp) does
not vanish for the set of values considered. Under these conditions
there exists one and only one system of solutions of the equations (45)
of the form
where fa, fa, • • • , <£y, are power series in xl} x2, • • • , a*, tvhich vanish
when xl = a;2 ••= • • • = xq = 0.
In order to simplify the notation, we shall restrict ourselves to
the case of two equations between two dependent variables u and v
and three independent variables x, y, and z :
/^CN (Fl = au -f bv + ex +dy + ez -\ ---- = 0,
(4o) <
I F2 = a'u + Vv + c'x + d'y + e'z ^ ---- = 0.
Since the determinant ab' — ba' is not zero, by hypothesis, the two
equations (46) may be replaced by two equations of the form
IX, §188] REAL ANALYTIC FUNCTIONS 403
mnpqr
(v = ^bmnpqr
where the left-hand sides contain no constant terms and no terms
of the first degree in u and v. It is easy to show, as above, that
these equations may be satisfied formally by replacing u and v by
power series in x, //, and z :
(48) u = 2c,.t,a;V*S v = 2c'iklx{ yk z1 ,
where the coefficients cikl and c\kl may be calculated from amnpqr and
bmn ( r by means of additions and multiplications only. In order to
show that these series converge, we need merely compare them with
the analogous expansions obtained by solving the two auxiliary
equations
where M, r, and p are positive numbers whose meaning has been
explained above. These two auxiliary equations reduce to a single
equation of the second degree
x + if + z
,r. |
_
2P + 4Af 2p + 4M 1 _ x + y + z
r
which has a single root which vanishes for x = y = z = 0, namely :
__
~
p2 ___ p2
~
4(p + 2M) 4(P -f 2M)
r
where a = r [p/(p + 4M)]2.
This root may be developed in a convergent power series when
ever the absolute values of x, y, and z are all less than or equal to
a/3. Hence the series (48) converges under the same conditions.
Let «x and vt be the solutions of (47) which are developable in
series. If we set u = ?/t + u', v = v^ -f v' in (47) and arrange the
result according to powers of x, y, z, u', v', each of the terms must
be divisible by u' or by v'. Hence, returning to the original varia
bles x, y, z, u, v, the given equations may be written in the form
M-«/ +(»-»iH =0,
404 SPECIAL SERIES [IX, § 189
where /, <f>, ft , ^ are power series in x, y, z, u, and v. In this
form the solutions u = ul} v =• v-i are exhibited. It is evident also
that no other solutions of (47') exist which vanish for x = y = z = 0.
For any other set of solutions must cause ffa — <&/\ to vanish,
and a comparison of (47) with (47') shows that the constant term
is unity in both / and <j>l} whereas the constant term is zero in
both/! and <£; hence the condition ffa — <£/i = 0 cannot be met by
replacing u and v by functions which vanish when x — y = z = 0.
189. Lagrange's formula. Let us consider the equation
(49) y = a + x$(y) ,
where <j>(y) is a function which is developable in a power series in y — a,
4>(y) = <t>(a) + (y - a) 0'(a) + (y ~ a)2 0"(a) + • • • ,
1 . £l
which converges whenever y — a does not exceed a certain number. By the
general theorem of § 187, this equation has one and only one root which
approaches a as x approaches zero, and this root is represented for sufficiently
small values of x by a convergent power series
y = a + aix -f a2x2 + • • •.
In general, if f(y) is a function which is developable according to positive
powers of y — a, an expansion of f(y) according to powers of x may be obtained
by replacing y by the development just found,
(50) f(y) =f(a) + Atx + A2x* + • • • + A,,x» + • . . ,
and this expansion holds for all values of x between certain limits.
The purpose of Lagrange's formula is to determine the coefficients
AI, A2, • • -, An, •••
in terms of a. It will be noticed that this problem does not differ essentially
from the general problem. The coefficient An is equal to the nth derivative of
f(y) for y = 0, except for a constant factor n!, where y is defined by (49); and
this derivative can be calculated by the usual rules. The calculation appears to
be very complicated, but it may be substantially shortened by applying the fol
lowing remarks of Laplace (cf. Ex. 8, Chapter II). The partial derivatives of
the function y defined by (49), with respect to the variables x and a, are given
by the formulae
whence we find immediately
i*.i\ du
(51)
ox da
where u =f(y). On the other hand, it is easy to show that the formula
da
— ! = -!><*) -
acJ dx I W c>a
IX, §189] REAL ANALYTIC FUNCTIONS 405
is identically satisfied, where F(y) is an arbitrary function of y. For either
side becomes
on performing the indicated differentiations. We shall now prove the formula
for any value of n. It holds, by (51), for n = 1. In order to prove it in gen
eral, let us assume that it holds for a certain number n. Then we shall have
dx» + l dan~ldx
But we also have, from (51) and (51'),
du
£l- £ [*«•£]-£[>*»•
caJ ca L cxj ca L
whence the preceding formula reduces to the form
cn + 1 u Bn
which shows that the formula in question holds for all values of n.
Now if we set x = 0, y reduces to a, u to /(a), and the nth derivative of u
with respect to x is given by the formula
da"-1
Hence the development of f(y) by Taylor's series becomes
/(2/)=/(a) + z0(a)/'(a) .
(52)
This is the noted formula due to Lagrange. It gives an expression for the
root y which approaches zero as x approaches zero. We shall find later the
limits between which this formula is applicable.
Note. It follows from the general theorem that the root y, considered as a
function of x and a, may be represented as a double series arranged according
to powers of x and a. This series can be obtained by replacing each of the
coefficients An by its development in powers of a. Hence the series (52) may
be differentiated term by term with respect to a.
Examples. 1) The equation
(53) y = a + ?(y*~l)
40G SPECIAL SERIES [IX, § 190
has one root which is equal to a when x = 0. Lagrange's formula gives the
following development for that root :
. 2 V 27 da
(54)
On the other hand, the equation (53) may be solved directly, and its roots are
, _J_M" d«-'(q2-l)n
1.2...nV27 do--»
y = - ± - Vl — 2ax + x2 .
The root which is equal to a when x = 0 is that given by taking the sign ~~.
Differentiating both sides of (54) with respect to a, we obtain a formula which
differs from the formula (32) of § 184 only in notation.
2) Kepler's equation for the eccentric anomaly u,*
(55) u = a + e sin u ,
which occurs in Astronomy, has a root u which is equal to a f or e = 0. Lagrange's
formula gives the development of this root near e = 0 in the form
(56) u = a e"
1.2 da 1.2..-n da»-i
Laplace was the first to show, by a profound process of reasoning, that this
series converges whenever e is less than the limit 0.662743 • •
190. Inversion. Let us consider a series of the form
(57) y = aix + azx2 + • • • + anxn + • • •,
where ai is different from zero and where the interval of convergence is( — r, + r).
If y be taken as the independent variable and x be thought of as a function of y,
by the general theorem of § 187 the equation (57) has one and only one root which
approaches zero with y, and this root can be developed in a power series in y :
(58) x = biy -f b2y2 + b^y3 + • • • + bny" + • • •.
The coefficients bi, b2, 63, • • • may be determined successively by replacing x in
(57) by this expansion and then equating the coefficients of like powers of y.
The values thus found are
bi = — , &2 = ~» bs = — - — 1 3, •••.
<*! a? af
The value of the coefficient bn of the general term may be obtained from
Lagrange's formula. For, setting
^(X) = tti + a2X + • • • + OnXn~l + • • •,
the equation (57) may be written in the form
1
*See p. 248, Ex. 19; and ZIWET, Elements of T?teoretical Mechanics, 2d ed.,
p. 356. — TRANS.
IX, §l!ii] REAL ANALYTIC FUNCTIONS 407
and the development of the root of this equation which approaches zero with y
is given by Lagrange's formula in the form
1 ?/" d»-i / 1 \"
+ + "
where the subscript 0 indicates that we are to set x = 0 after performing the
indicated differentiations.
The problem just treated has sometimes been called the reversion of series.
191. Analytic functions. In the future we shall say that a func
tion of any number of variables x, y, z, ••• is analytic if it can be
developed, for values of the variables near the point x0, y0, z0, •••,
in a power series arranged according to increasing powers of
x ~ xo> V — 2/o j z ~ zo> "' which converges for sufficiently small
values of the differences x — x0, • • • . The values which ar0, y0, z0, • • •
may take on may be restricted by certain conditions, but we shall
not go into the matter further here. The developments of the pres
ent chapter make clear that such functions are, so to speak, inter
related. Given one or more analytic functions, the operations of
integration and differentiation, the algebraic operations of multipli
cation, division, substitution, etc., lead to new analytic functions.
Likewise, the solution of equations whose left-hand member is ana
lytic leads to analytic functions. Since the very simplest functions,
such as polynomials, the exponential function, the trigonometric
functions, etc., are analytic, it is easy to see why the first functions
studied by mathematicians were analytic. These functions are still
predominant in the theory of functions of a complex variable and in
the study of differential equations. Nevertheless, despite the funda
mental importance of analytic functions, it must not be forgotten
that they actually constitute merely a very particular group among
the whole assemblage of continuous functions.*
192. Plane curves. Let us consider an arc AB of a plane curve.
We shall say that the curve is analytic along the arc AB if the
coordinates of any point M which lies in the neighborhood of any
fixed point M0 of that arc can be developed in power series arranged
according to powers of a parameter t — t0,
(59) T" -rw-^o + aiC to)+*2(t toy + ... + an(t
y = f(f) = y0 + bl(t- f0) + bz (t - toy + • • • + bn(t -
which converge for sufficiently small values of t — t0.
* In the second volume an example of a non-analytic function will be given, all of
whose derivatives exist throughout an interval (a, b).
408 SPECIAL SERIES [IX, § 192
A point 3/o will be called an ordinary point if in the neighbor
hood of that point one of the differences y — y0, x — x0 can be
represented as a convergent power series in powers of the other.
If, for example, y — y0 can be developed in a power series in
x x0)
(60) y — y(> = ^(x — x0) + ca (x — x0)2 H ----- 1- cn(x — z0)n H ---- ,
for all values of x between xn — h and -a-0 + h, the point (x0 , y0~) is
an ordinary point. It is easy to replace the equation (60) by two
equations of the form (59), for we need only set
(61)
If cv is different from zero, which is the case in general, the equa
tion (60) may be solved for x — x0 in a power series in y — y0 which
is valid whenever y — y0 is sufficiently small. In this case each of
the differences x — x0, y — y0 can be represented as a convergent
power series in powers of the other. This ceases to be true if ct is
zero, that is to say, if the tangent to the curve is parallel to the
x axis. In that case, as we shall see presently, x — x0 may be devel
oped in a series arranged according to fractional powers of y — y0.
It is evident also that at a point where the tangent is parallel to
the y axis x — x0 can be developed in power series in y — yoy but
y — t/o cannot be developed in power series in x — x0.
If the coordinates (a-, ?/) of a point on the curve are given by the
equations (59) near a point M0, that point is an ordinary point if
at least one of the coefficients al} bi is different from zero.* If al
is not zero, for example, the first equation can be solved for t — t0
in powers of x — x0, and the second equation becomes an expansion
of y — y0 in powers of x — x0 when this solution is substituted for
t-t0.
The appearance of a curve at an ordinary point is either the cus
tomary appearance or else that of a point of inflection. Any point
which is not an ordinary point is called a singular point. If all
the points of an arc of an analytic curve are ordinary points, the
arc is said to be regular.
* This condition is sufficient, but not necessary. However, the equations of any
curve, near an ordinary point M0, may always be written in such a way that 04 and
bl do not both vanish, by a suitable choice of the parameter. For this is actually
accomplished in equations (61). See also second footnote, p. 409. —TRANS.
IX, §193] KEAL ANALYTIC FUNCTIONS 409
If each of the coefficients a± and bv is zero, but a2, for example,
is different from zero, the first of equations (59) may be written in
the form (x — xu)* = (t — £0) [«2 -f a8 (t — £„) -\ ---- ]*, where the right-
hand member is developable according to powers of t — t0. Hence
t — t0 is developable in powers of (x — a^)*, and if t — t0 in the
second equation of (59) be replaced by that development, we obtain
a development for y — y0 in powers of (x — z0)* :
In this case the point (z0, y0~) is usually a cusp of the first kind.*
The argument just given is general. If the development of
x — x0 in powers of t — t0 begins with a term of degree n, y — y0
can be developed according to powers of (x — a-0). The appearance
of a curve given by the equation (59) near a point (x0, y0) is of
one of four types : a point with none of these peculiarities, a point
of inflection, a cusp of the first kind, or a cusp of the second kind.*
193. Skew curves. A skew curve is said to be analytic along an arc
AB if the coordinates x, y, z of a variable point M can be developed
in power series arranged according to powers of a parameter t — t0
f x = x0 + ai(t - *„) H ----- h an(t - tu)n -i ---- ,
(62) J y = y0 + b, (t - *0) + • • • + bn (t - t,}n + -..,
(*SSK9+Cl(t-tj + -~+ Cn(t - t0)n + • • -,
in the neighborhood of any fixed point M0 of the arc. A point
M0 is said to be an ordinary point if two of the three differences
x — xoi y — y<» z — so can be developed in power series arranged
according to powers of the third.
It can be shown, as in the preceding paragraph, that the point
M0 will surely be an ordinary point if not all three of the coefficients
al,bl, cl vanish. Hence the value of the parameter t for a singular
point must satisfy the equations f
^ = 0 dy_Q dz_0
dt dt U' dt
* For a cusp of the first kind the tangent lies between the two branches. For a
cusp of the second kind both branches lie on the same side of the tangent. The
point is an ordinary point, of course, if the coefficients of the fractional powers
happen to be all zeros. — TRANS.
t These conditions are not sufficient to make the point 3f0 , which corresponds to
a value t0 of the parameter, a singular point when a point M of the curve near Jf0
corresponds to several values of t which approach t0 as M approaches M0 . Such is
the case, for example, at the origin on the curve defined by the equations x = t3 ,
410 SPECIAL SERIES [IX, § 194
Let x0, y0, z0 be the coordinates of a point M0 on a skew curve T
whose equations are given in the form
(63) F(x,y,z)=0, F,(x, </,*)= 0,
where the functions Fand Ft are power series in x — x0, y — y0, z — z0.
The point M0 will surely be an ordinary point if not all three of
the functional determinants
D(F, F,} £(F, FQ D(F, FQ
D(x, y) ' D(y, z) ' D(z, x)
vanish simultaneously at the point x = x0, y = ?/0, z = z0. For if
the determinant D(F, Fi)/D(x, y), for example, does not vanish at
M0, the equations (63) can be solved, by § 188, for x — XQ and y — yQ
as power series in z — z0.
194. Surfaces. A surface S will be said to be analytic throughout
a certain region if the coordinates x, y, z of any variable point M
can be expressed as double power series in terms of two variable
parameters t — t0 and u — HO
ex — x0 = a10(t — f0) + «oi(M - OH ,
(64) J y - y0 = b10(t - *0) + b01(u - w0) -\ ,
U - «0 = cio(t — <o) + coi (u — ?'o) H 1
in the neighborhood of any fixed point M0 of that region, where
the three series converge for sufficiently small values of t — 10 and
u — u0. A point M0 of the surface will be said to be an ordinary
point if one of the three differences x — x0, y — y0, z — z0 can be
expressed as a power series in terms of the other two. Every point
M0 for which not all three of the determinants
D(y, g) D(z, x)^ D(x, y}
D(t, u)' D(t, u)' D(t, u)
vanish simultaneously is surely an ordinary point. If, for exam
ple, the first of these determinants does not vanish, the last two of
the equations (64) can be solved for t — t0 and u — u0, and the first
equation becomes an expansion of x — x0 in terms of y — y0 and
z — z0 upon replacing t — t0 and u — u0 by these values.
Let the surface S be given by means of an unsolved equation
F(x, y, z) = 0, and let x0, y0, z0 be the coordinates of a point M0
of the surface. If the function F(x, y, z) is a power series in
x — xo> y — y<» z — zo) an(i ^ n°t all three of the partial derivatives
dF/dx0, 8F/8y0, 8F/dz0 vanish simultaneously, the point 3/0 is surely
an ordinary point, by § 188.
IX,§195J TRIGONOMETRIC SERIES 411
Note. The definition of an ordinary point on a curve or on a sur
face is independent of the choice of axes. For, let 3/0 (x0, y0, «0) be an
ordinary point on a surface S. Then the coordinates of any neigh
boring point can* be written in the form (64), where not all three of
the determinants D(y, z)/D(t, u), D(z, x)/D(t, u}, D(x, y)/D(t, it}
vanish simultaneously for t = t0, u = u0. Let us now select any new
axes whatever and let
A' = aix + fay + yiz + 81}
Y= a^x + {32y -f yzz + S2,
Z = a3x + fay + y3z + 83
be the transformation which carries x, y, z into the new coordinates
X, Y, Z, where the determinant A = D(X, Y, Z)/D(x, y, z) is differ
ent from zero. Replacing x, y, z by their developments in series
(64), we obtain three analogous developments for X, Y, Z ; and we
cannot have
1>(X, Y} = Z>( J^Z) = D(Z, X} =
J)(t, if) ' ' D(t, u) ~ D(t, u)
for t = £0, u = ii0, since the transformation can be written in the form
X = A1X + B1Y+ C\Z + Dly
y = A2X + B2 Y + C2Z + J92,
z = A3X + B3 Y+C3Z + Dt,
and the three functional determinants involving .Y, Y, Z cannot
vanish simultaneously unless the three involving x, y, z, also vanish
simultaneously.
IV. TRIGONOMETRIC SERIES MISCELLANEOUS SERIES
195. Calculation of the coefficients. The series which we shall study
in this section are entirely different from those studied above.
Trigonometric series appear to have been first studied by D. Ber
noulli, in connection with the problem of the stretched string. The
process for determining the coefficients, which we are about to give,
is due to Euler.
Let /(a:) be a function defined in the interval (a, &). We shall
first suppose that a and b have the values — TT and + TT, respec
tively, which is always allowable, since the substitution
2-7TX — (a -f 1} 7T
x' = r-*-
o — a
* See footnote, p. 408. —TRANS.
412 SPECIAL SERIES [IX, § 195
reduces any case to the preceding. Then if the equation
(tn
(65) /(#) = -^ -f («i cos x + b: sin x) -\ \- (amcos mx + &msin mx) -\
holds for all values of x between — TT and + TT, where the coefficients
ao) ai> bi> • • •) am> bm>--- are unknown constants, the following device
enables us to determine those constants. We shall first write down
for reference the following formulae, which were established above,
for positive integral values of m and n :
(66)
sin mx dx = 0 ;
L
L
f
\J — 71
/cos'-mxdx = I — dx = TT, if m 3= 0;
7T V/-7T ^
r + "
I sin mx sin nx dx
\J — 7T
f + * cos (m — n) x — cos (m -f n)x ^ _
— / o "» x
C/-7T
cos m« rfa; = 0, if m ^ 0 ;
r
cos wcc cos nx dx
" cos(m — ri)x + cos (m + ri)x 7 .„
— ^ * — 5 — — rfx = 0 , if m =£ n ;
2
r + 7r . r + " 1 — cos 2mrc , .„
I sm2ma;rfx= I ^ ^x =TT, ifw^O;
I
sin mx cos nx dx
X77 sin (m + n) x + sin (TO — n)x __
2
Integrating both sides of (65) between the limits — TT and -f TT,
the right-hand side being integrated term by term, we find
/"» -\- 7T /^ "t" T
I /(x) dx = -£ I dx = 7ra0,
J-TT ««/-«
which gives the value of a0. Performing the same operations upon
the equation (65) after having multiplied both sides either by cos mx
IX, §195] TRIGONOMETRIC SERIES 413
or by sin mx, the only term on the right whose integral between — TT
and + TT is different from zero is the one in cos2 mx or in sin2 mx.
Hence we find the formulae
/ + "• (* + •*
f(x) cos mx dx = 7nzm, I f(x) sin mx dx = 7rbm,
TT \J — TT
respectively. The values of the coefficients may be assembled as
follows :
(67)
a0 = - I /(a) da, am = - I /(a) cos ma da,
** t/ — n *i^/ — jr
i r+)r
bm = — I f(a~) sin mar c?nr.
T7" t/-n-
The preceding calculation is merely formal, and therefore tenta
tive. For we have assumed that the function f(x) can be developed
in the form (65), and that that development converges uniformly
between the limits — TT and -f TT. Since there is nothing to prove,
a priori, that these assumptions are justifiable, it is essential that
we investigate whether the series thus obtained converges or not.
Eeplacing the coefficients a{ and b{ by their values from (67) and
simplifying, the sum of the first (m + 1) terms is seen to be
1 f+7r fl 1
Sm+l = — I /(«) -^ + cos(a— a-) + cos2(rt— x}-\ ----- \-COSm(a— x) \da.
But by a well-known trigonometric formula we have
2m +1
sin — - — a
- + cos a -f cos 2a + • • • + cos m a = - - - >
2 . a
2sm-
whence
. 2m + 1
sin — - — (a — a-)
or, setting a = x -f-
(68) S^
The whole question is reduced to that of finding the limit of this
sum as the integer m increases indefinitely. In order to study this
question, we shall assume that the function f(x) satisfies the fol
lowing conditions :
414 SPECIAL SERIES [IX, § 196
1) The function f(x) shall be in general continuous between — TT
and + TT, except for a, finite number of values of x, for which its value
may change suddenly in the following manner. Let c be a number
between — TT and + TT. For any value of c a number h can be found
such that f(x) is continuous between c — h and c and also between
c and c + h. As c approaches zero, f(c -f- c) approaches a limit which
we shall call f(c + 0). Likewise, f(c — e) approaches a limit which
we shall call f(c — 0) as c approaches zero. If the function f(x)
is continuous for x = c, we shall have /(c) —f(c + 0) =/(c — 0). If
f(c + 0) =£ f(c — 0), f(x) is discontinuous for x = c, and we shall agree
to take the arithmetic mean of these values [/(c + 0) + f(c — 0)]/2
for /(c). It is evident that this definition of /(c) holds also at points
where f(x) is continuous. We shall further suppose that /(— TT + e)
and /(TT — e) approach limits, which we shall call /(— TT + 0) and
/(TT — 0), respectively, as e approaches zero through positive values.
The curve whose equation is y —f(x) must be similar to that of
Fig. 11 on page 160, if there are any discontinuities. We have
already seen that the function /(x) is integrable in the interval from
— TT to + TT, and it is evident that the same is true for the product
of f(x) by any function which is continuous in the same interval.
2) It shall be possible to divide the interval (— TT, + TT) into a
finite number of subintervals in such a way thaty(cr) is a monoton-
ically increasing or a monotonically decreasing function in each of
the subintervals.
For brevity we shall say that the function f(x) satisfies Dirichlet's
conditions in the interval (— TT, + TT). It is clear that a function
which is continuous in the interval (— TT, + TT) and which has a
finite number of maxima and minima in that interval, satisfies
Dirichlet's conditions.
196. The integral /h f (x) [sin nx/sin x] dx. The expression obtained
for <STO + 1 leads us to seek the limit of the definite integral
C sin nx
f(x) — dx
Jo since
as n becomes infinite. The first rigorous discussion of this ques
tion was given by Lejeune-Dirichlet.* The method which we shall
employ is essentially the same as that given by Bonnet. t
* Crelle's Journal, Vol. IV, 1829.
t Mtmoires des savants etrangers publics par 1' Academic de Belgique, Vol. XXIII.
IX, §196] TRIGONOMETRIC SERIES 415
Let us first consider the integral
r>h
I . . sin nx ,
(69) ./ = I <K*)~ — dx>
c/O
where h is a positive number less than TT, and <j>(x) is a function
which satisfies Dirichlet's conditions in the interval (0, A). If <f>(x)
is a constant C, it is easy to find the limit of /. For, setting y = nx,
we may write
sink
I sin y 7
J=C\ —2-dy,
Jo y
and the limit of J as n becomes infinite is CTr/2, by (39), § 176.
Next suppose that <f>(x) is a positive monotonically decreasing
function in the interval (0, h). The integrand changes sign for
all values of x of the form kir/n. Hence J may be written
j= Uo - Ul + u2 - u3 + • • • + (- 1)*%- + ••• + (- I)m0",«, 0 < 0 < 1,
where
C "
I
Jk*
sn nx .
- dx
and where the upper limit h is supposed to lie between mir/n and
(ra -f T)7r/n. Each of the integrals uk is less than the preceding.
For, if we set nx = &TT + y in uk, we find
y + kw\ sin y
n / y + A:TT *
and it is evident, by the hypotheses regarding <£(#), that this inte
gral decreases as the subscript k increases. Hence we shall have
the equations
./ = UQ — \Ml — ^"i) y^'s ^4/ ' ' ')
which show that / lies between ?/0 and n0 — u^ . It follows that / is
a positive number less than ii0, that is to say, less than the integral
/
Jo
smnx
- -dx.
x
But this integral is itself less than the integral
Jo y
where A denotes the value of the definite integral //[(sin y)/y~\ dy.
416 SPECIAL SERIES [IX, § 196
The same argument shows that the definite integral
Xh
, . ^ sm nx .
*(x)- -dx,
x
where c is any positive number less than h, approaches zero as n
becomes infinite. If c lies between (i — l)?r/n and i-rr/n, it can be
shown as above that the absolute value of J' is less than
C " . sin nx , C
<p(x) dx +
Jc X Jiz
n
and hence, a fortiori, less than
snnx
_
c \ n
Hence the integral approaches zero as n becomes infinite.*
This method gives us no information if c = 0. In order to dis
cover the limit of the integral J, let c be a number between 0
and h, such that <f>(x) is continuous from 0 to c, and let us set
<l>(x) = <f>(c) + \f/(x"). Then i^(a-) is positive and decreases in the
interval (0, c) from the value <£(+ 0) - <£(c) when x = 0 to the
value zero when x = c. If we write / in the form
/""sin nx Cr sin nx C
J = <£(c) I — — dx + I $(x) - dx +
JO X Jo X Jc
and then subtract (7r/2)<Ti(-f 0), we find
sin nx 7
-- dx
(70)
/•e . ,-> .
I sinnx I smnir
I f(x) - -dx+ <#>(x) -
JO -^ Jc x
In order to prove that ./ approaches the limit (Tr/2) ^>(+ 0), it will
be sufficient to show that a number m exists such that the absolute
* This result may be obtained even more simply by the use of the second theorem
of the mean for integrals (§75). Since the function <f>(r) is a decreasing function,
that formula gives
. . sin nx , 0(c) r 1 <f>M
(«) — — ax = -J- J sin wx dx = - -j* (cos nc - cosn|) ,
and the right-hand member evidently approaches zero.
IX, §196] TRIGONOMETRIC SERIES 417
value of each of the terms on the right is less than a preassigned
positive number e/4 when n is greater than m. By the remark
made above, the absolute value of the integral
/'
Jo
sin nx
ax
is less than A$(+ 0) = A [<£(+ 0) — <£(e)]. Since <£(z) approaches
<£(+ 0) as x approaches zero, c may be taken so near to zero that
A [>(+ 0) - £(c)] and (7r/2)[>(+ 0) - «£(c)] are both less than e/4.
The number c having been chosen in this way, the other two terms
on the right-hand side of equation (70) both approach zero as n
becomes infinite. Hence n may be chosen so large that the abso
lute value of either of them is less than e/4. It follows that
(71) lim/=|*(+0).
»(=QO A
We shall now proceed to remove the various restrictions which
have been placed upon <f>(x) ^n the preceding argument. If <f>(x) is
a monotonically decreasing function, but is not always positive, the
function \j/(x) — <f>(x) + C is a positive monotonically decreasing func
tion from 0 to h if the constant C be suitably chosen. Then the
formula (71) applies to \j/(x). Moreover we may write
(* * sinnx Ck sinnx C
I £(*) — —dx= <K*)- -dx-C
JO Jo JO
h sinnx
— dx,
and the right-hand side approaches the limit (w/2) \j/(-{- 0) — (?r/2) C,
that is, (?r/2) <#»(+ 0).
If <£(#) is a monotonically increasing function from 0 to h, — <f>(x~)
is a monotonically decreasing function, and we shall have
Ch sin nx C
I ^(*) - - dx = - I
Jo * t7o
sin nx
— - dx.
Hence the integral approaches (7r/2)<£(+ 0) in this case also.
Finally, suppose that <£(#) is any function which satisfies Dirich-
let's conditions in the interval (0, A). Then the interval (0, A)
may be divided into a finite number of subintervals (0, a), (a, b),
(b, c), • • -, (7, A), in each of which <f>(x~) is a monotonically increasing
or decreasing function. The integral from 0 to a approaches the limit
(7T/2) <£(-)- 0). Each of the other integrals, which are of the type
»=/'
Ja
sinnx
dx,
418 SPECIAL SERIES [IX, § 107
approaches zero. For if <f>(x~) is a monotonically increasing function,
for instance, from a to b, an auxiliary function \j/(x) can be formed
in an infinite variety of ways, which increases monotonically from
0 to b, is continuous from 0 to a, and coincides with <f>(x) from a to b.
Then each of the integrals
C'
/
t/O
sinnx sinnx
— —<**,
approaches $(+ 0) as n becomes infinite. Hence their difference,
which is precisely H, approaches zero. It follows that the formula
(71) holds for any function <f>(x) which satisfies Dirichlet's condi
tions in the interval (0, h).
Let us now consider the integral
(72) f=f(x)dX) (X/K7T,
where f(x) is a positive monotonically increasing function from
0 to A. This integral may be written
C v ff \ x
= 1 f(x)~ —
Jo L ^ ' sm x
n s
-,
dx,
sill x J x
and the function tf>(x) = f(x) x/sin x is a positive monotonically
increasing function from 0 to h. Since /(+ 0) = <£(+ 0), it follows
that
(73) lim 7 = £/(+ 0).
This formula therefore holds if f(x) is a positive monotonically
increasing function from 0 to h. It can be shown by successive
steps, as above, that the restrictions upon f(x) can all be removed,
and that the formula holds for any function f(x) which satisfies
Dirichlet's conditions in the interval (0, li).
197. Fourier series. A trigonometric series whose coefficients are
given by the formulae (67) is usually called a Fourier series. Indeed
it was Fourier who first stated the theorem that any function arbi
trarily defined in an interval of length 2?r may be represented by a
series of that type. By an arbitrary function Fourier understood
a function which could be represented graphically by several cur
vilinear arcs of curves which are usually regarded as distinct curves.
We shall render this rather vague notion precise by restricting our
discussion to functions which satisfy Dirichlet's conditions.
IX, §197]
TRIGONOMETRIC SERIES 419
In order to show that a function of this kind can be represented
by a Fourier series in the interval (— TT, + TT), we must find the
limit of the integral (68) as m becomes infinite. Let us divide
this integral into two integrals whose limits of integration are
0 and (TT — r)/2, and — (TT + x)/2 and 0, respectively, and let us
make the substitution y = — z in the second of these integrals.
Then the formula (68) becomes
sn
1 C * s, ox sin (2m + 1) * ,
I f(x — 2z) — — dz .
TT Jo J v sm «
When x lies between — TT and 4- TT, (TT — a;)/2 and (TT + #)/2 both
lie between 0 and TT. Hence by the last article the right-hand
side of the preceding formula approaches
as m becomes infinite. It follows that the series (65) converges and
that its sum is/(x) for every value of x between — TT and + TT.
Let us now suppose that x is equal to one of the limits of the
interval, — TT for example. Then Sm + l may be written in the form
1 C*., , 0 Xsin(2m4-1)//j
= - I /(— TT + 2y) -- i— ; — — dy
7rJ0 J smy
The first integral on the right approaches the limit /(— TT + 0)/2.
Setting y = TT — 2 in the second integral, it takes the form
sm*
which approaches /(TT — 0)/2. Hence the sum of the trigonometric
series is [/(TT — 0) +/(— TT + 0)] /2 when x = — TT. It is evident
that the sum of the series is the same when x = + TT.
If, instead of laying off x as a length along a straight line, we
lay it off as the length of an arc of a unit circle, counting in the
420 SPECIAL SERIES [IX, § 197
positive direction from the point of intersection of the circle with
the positive direction of some initial diameter, the sum of the series
at any point whatever will be the arithmetic mean of the two limits
approached by the sum of the series as each of the variable points
ra' and ra", taken on the circumference on opposite sides of ra,
approaches m. If the two limits /(— TT + 0) and f(ir — 0) are
different, the point of the circumference on the negative direction
of the initial line will be a point of discontinuity.
In conclusion, every function which is defined in the interval
(— TJ -f- "") and which satisfies Dirichlet's conditions in that inter
val may be represented by a Fourier series in the same interval.
More generally, let f(x) T^e a function which is defined in an
interval (a, a + 2?r) of length 2?r, and which satisfies Dirichlet's
conditions in that interval. It is evident that there exists one and
only one function F(x) which has the period 2?r and coincides with
/(#) in the interval (a, a + 2?r). This function is represented, for
all values of x, by the sum of a trigonometric series whose coeffi
cients am and bm are given by the formulas (67):
am = ~ I F(r) cos mx dx, bm = - I F(x) sin mx dx .
The coefficient am, for example, may be written in the form
1 r* i r«- 2*-
am — — I F(x) cos mx dx + - I F(x) cos mx dx,
*V«-tte vj_n
where a is supposed to lie between 2hir — TT and 2hjr + TT. Since
F(x} has the period 2?r and coincides with f(x) in the interval
(a, a + 2?r), this value may be rewritten in the form
( 1 /l2'"r + !r ^»a: + 2T
\am = ~ I f(y) cos mx dx + / /'(.r) cos mx dx
(74) I Ja Jm*+m
-• ,^a + 2jr
= — / f(x) cos mx dx .
V. t/ ft
Similarly, we should find
(75) bm = ~ I f(x) sin mx dx .
"ft J a
Whenever a function f(x) is defined in any interval of length 2?r,
the preceding formulae enable us to calculate the coefficients of its
development in a Fourier series without reducing the given interval
to the interval (— TT, + TT).
IX, §198] TRIGONOMETRIC SERIES 421
198. Examples. 1) Let us find a Fourier series whose sum is — 1 for
— it < x < 0, and +1 f or 0 < x < + it. The formulae (67) give the values
1 r° 1 cn
a0 = — I — dx H — I dx = 0 ,
It J-ir Tt Jo
1 r° i r*
am = — I — cos mxdx -\ — cos mx dx = 0 ,
It J -•n It JQ
I f° . j , ! T* • A 2 — cosmTT — cos(— mir)
nm = ~ — sm mx dx -\ — sin mx ax = - — > - '- .
71 J -T, Tt Jo mit
If m is even, bm is zero. If m is odd, bm is 4/mir. Multiplying all the coeffi
cients by 7T/4, we see that the sum of the series
(76) v = -i- Sin3x + +
1 3 2m + 1
is — 7T/4 for — it < x < 0, and + ff/4 for 0 < x < it. The point x = 0 is a point
of discontinuity, and the sum of the series is zero when x = 0, as it should be.
More generally the sum of the series (76) is 7t/4 when sin x is positive, — ?r/4
when sin x is negative, and zero when sin x — 0.
The curve represented by the equation (76) is composed of an infinite number
of segments of length n of the straight lines y = ± 7T/4 and an infinite num
ber of isolated points (y = 0, x = kit) on the x axis.
2) The coefficients of the Fourier development of x in the interval from 0 to
27T are
1 r-*
do = — I xdx = 2?r,
Tt Jo
1 r2" rxsinmx-l2* 1 c^ •
dm = -- x cos mxdx = -{ -- I sin mx dx = 0 ,
Tt Jo \_ 11171 Jo mit Jo
1 r2" rxcos?/ixl2;r 1 (**" 2
om = - x sin mx dx = — \ -\ --- | cos mx dx = ---
n Jo \_ mit Jo iwt Jo m
Hence the formula
x it sinx sin 2x sin3x
__
22
is valid for all values of x between 0 and 2?r. If we set y equal to the series on
the right, the resulting equation represents a curve composed of an infinite num
ber of segments of straight lines parallel to y — x/2 and an infinite number of
isolated points.
Note. If the function /(x) defined in the interval ( - Tt, + Tt) is even, that is
to say, if /(— x) =/(x), each of the coefficients bm is zero, since it is evident that
nO n-n
I f(x) sin mx dx — — I /(x) sin mx dx .
J-n Jo
Similarly, if /(x) is an odd function, that is, if /(— x) = — /(x), each of the
coefficients am is zero, including a0. A function /(x) which is defined only in
422 SPECIAL SERIES [IX, § 199
the interval from 0 to it may be defined in the interval from — it to 0 by either
of the equations
/(-*)= /(«) or /( - z) = - f(x)
if we choose to do so. Hence the given function /(x) may be represented either
by a series of cosines or by a series of sines, in the interval from 0 to ic.
199. Expansion of a continuous function. Weierstrass' theorem. Let /(z) be a
function which is defined and continuous in the interval (a, 6). The following
remarkable theorem was discovered by Weierstrass : Given any preassigned posi
tive number e, a polynomial P(x) can always be found such that the difference
f(x)—P(x) is less than e in absolute value for all values of x in the interval (a, 6).
Among the many proofs of this theorem, that due to Lebesgue is one of the
simplest.* Let us first consider a special function \j/(x) which is continuous in
the interval (—1, +1) and which is defined as follows : ^(x) = 0 for — 1 < x < 0,
V'(x) = 2kx for 0 < x < 1, where fc is a given constant. Then \f/(x) = (x -f | x |) k.
Moreover for - 1 < x < + 1 we shall have
and for the same values of x the radical can be developed in a uniformly con
vergent series arranged according to powers of (1 - x2). It follows that x|, and
hence also i//(x), may be represented to any desired degree of approximation in
the interval (— 1, + 1) by a polynomial.
Let us now consider any function whatever, /(x), which is continuous in
the interval (a, 6), and let us divide that interval into a suite of subintervals
(<*oi ai), (a\ , a2), • • • , (a,,_i , an), where a = a0 < ax < a2 < • • • < a,,_i < an = 6,
in such a way that the oscillation of /(x) in any one of the subintervals is less
than e/2. Let L be the broken line formed by connecting the points of the
curve y =f(x) whose abscissae are a0, ax, a2, • • •, b. The ordinate of any point
on L is evidently a continuous function 0(x), and the difference /(x) - <f>(x) is
less than e/2 in absolute value. For in the interval (aM_!, aM), for example,
we shall have
/(x) - 0(x) = [/(x) -/(<v_,)] (1 - 6) + [/(x) -/(aM)] 0,
where x - aM_i = 0(aM - a^_i). Since the factor d is positive and less than
unity, the absolute value of the difference/ — <f> is less than e(l - 6 + e)/2 = e/2.
The function </>(x) can be split up into a sum of n functions of the same type as
V'(x). For, let AQ, AI, A2, • • • , An be the successive vertices of L. Then </>(x)
is equal to the continuous function ^ (x) which is represented throughout the
interval (a, 6) by the straight line A0Al extended, plus a function 0i(x) which
is represented by a broken line A'0A{ • • • A'» whose first side A'QA\ lies on the
x axis and whose other sides are readily constructed from the sides of L. Again,
the function fa (x) is equal to the sum of two functions ^2 and 02 , where f2 is
zero between a0 and cti, and is represented by the straight line A\A'i extended
between aj and 6, while 02 is represented by a broken line A(,'Ai'A'2'- • • A'n' whose
first three vertices lie on the x axis. Finally, we shall obtain the equation
0 = f i + </"2 + • • • + </-„ , where ^ is a continuous function which vanishes
between a0 and o,-_i and which is represented by a segment of a straight line
* Bulletin des sciences mathtmatiques, p. 278, 1898.
IX, §200] TRIGONOMETRIC SERIES 423
between ai_i and b. If we then make the substitution X = mx + n, where m
and n are suitably chosen numbers, the function \j/i(x) may be defined in the
interval (— 1, + 1) by the equation
and hence it can be represented by a polynomial with any desired degree of
approximation. Since each of the functions f;(x) can be represented in the
interval (a, 6) by a polynomial with an error less than c/2n, it is evident that the
sum of these polynomials will differ from /(x) by less than e.
It follows from the preceding theorem that any function f(x) which is contin
uous in an interval (a, b) may be represented by an infinite series of polynomials
which converges uniformly in that interval. For, let «i , c2 , • • • , e« , • • • be a sequence
of positive numbers, each of which is less than the preceding, where en approaches
zero as n becomes infinite. By the preceding theorem, corresponding to each of
the e's a polynomial P»(x) can be found such that the difference /(z) — P,(x) is
less than e,- in absolute value throughout the interval (a, b). Then the series
^1 (3) + tP2 (X) - Pi (X)] + • • • + [Pn (X) - P.-i (»)] + • • •
converges, and its sum is/(x) for any value of x inside the interval (a, b). For
the sum of the first n terms is equal to PB(x), and the difference /(x) — <SM, which
is less than e,, , approaches zero as n becomes infinite. Moreover the series con
verges uniformly, since the absolute value of the difference /(x) — Sn will be less
than any preassigned positive number for all values of n which exceed a certain
fixed integer N, when x has any value whatever between a and b.
200. A continuous function without a derivative. We shall conclude this chapter
by giving an example due to Weierstrass of a continuous function which does
not possess a derivative for any value of the variable whatever. Let 6 be a posi
tive constant less than unity and let a be an odd integer. Then the function
F(x) defined by the convergent infinite series
-r ^
(78) F(x) = V b" cos (a" nx)
is continuous for all values of x, since the series converges uniformly in any
interval whatever. If the product ab is less than unity, the same statements
hold for the series obtained by term-by-term differentiation. Hence the func
tion F(x) possesses a derivative which is itself a continuous function. We shall
now show that the state of affairs is essentially different if the product ab exceeds
a certain limit.
In the first place, setting
m-l
Sm = - ^ b" (cos [an TT(Z + h)] — cos (an TTX) } ,
H=U
R,n — ^ b" {cos [a" TT(X + h)] — cos (a" nx)} ,
we may write
(79) ffl±*Li!fiL>. «.+*..
h
SPECIAL SERIES [IX, §200
On the other hand, it is easy to show, by applying the law of the mean to the
function cos(a"7rz), that the difference cos[a"7r(z + h)] — cos(anitx) is less than
nan | h \ in absolute value. Hence the absolute value of Sm is less than
m-l
anbn = n
ab-l
n=(l
and consequently also less than 7r(ab)m/(ab — 1), if ab >1. Let us try to find a
lower limit of the absolute value of Bm when h is assigned a particular value.
We shall always have
amx = am+ £,„,
where am is an integer and £m lies between — 1/2 and + 1/2. If we set
p — ?
/j _ 5S ™,
a"1
where em is equal to ± 1, it is evident that the sign of h is the same aa that of
em, and that the absolute value of h is less than 3/2am. Having chosen h in this
way, we shall have
a"7f(x + h) — an-'nam7c(x + h) = an-mn(am + em).
Since a is odd and em = ± 1, the product an-'»(am -f em) is even or odd with
am + 1, and hence
cos[a"7r(z + A)] = (— l)am + l.
Moreover we shall have
coa(an7tx) = cos(a"-mam7rz) = cos[a"-m*(am + £m)]
= cos (a" - m am Tt ) cos (a" - "' £m n) ,
or, since a"~mam is even or odd with am,
CQS (fin 7f%\ ' — ( 1 ^""* COS {an — m ^ 7T^
It follows that we may write
(_1)g, + i *'
"m —
n
Since every term of the series is positive, its sum is greater than the first term, and
consequently it is greater than 6m since £,„ lies between — 1/2 and + 1/2. Hence
or, since \h\< 3/2am,
\
If a and b satisfy the inequality
(80)
we shall have
(80) a6>l + — ,
•
3' ab-l
whence, by (79),
F(x + h)- F(x)
> |B«|-|Sn
2
3 x a6-l
IX, Exs.] EXERCISES 425
As m becomes infinite the expression on the extreme right increases indefinitely,
while the absolute value of h approaches zero. Consequently, no matter how
small e be chosen, an increment h can be found which is less than « in abso
lute value, and for which the absolute value of [F(x + h) — F(x)~\/h exceeds any
preassigned number whatever. It follows that if a and b satisfy the relation (80),
the function F(x) possesses no derivative for any value of x whatever.
EXERCISES
1. Apply Lagrange's formula to derive a development in powers of x of that
root of the equation y'2 = ay + x which is equal to a when x = 0.
2. Solve the similar problem for the equation y — a + xym + l = 0. Apply the
result to the quadratic equation a — bx + ex2 = 0. Develop in powers of c that
root of the quadratic which approaches a/6 as c approaches zero.
3. Derive the formula
l+x \ I
4. Show that the formula
Vl + x 1 + x 2 \1 + x/ 2.4
holds whenever x is greater than — 1/2.
5. Show that the equation
c
+
2 1 + x2 2 . 4 \1 + x2/ 2 . 4 . G \1 + x2
holds for values of x less than 1 in absolute value. What is the sum of the series
when | x | > 1 ?
6. Derive the formula
1.2 a + * 1-2.3
7. Show that the branches of the function sinmx and cosmx which reduce
to 0 and 1, respectively, when sinx = 0 are developable in series according 10
powers of sin x :
f . m2 - 1 . (m2 - l)(m2 - 0) . n
sm mx = m sm x smsx + — sm5x — • • • ,
L 1.2.3 1.2.3.4.5 J
?n2 . m2(m2-4) .
cos mx = 1 sin2x H ' sm4x — • • •.
1.2 1.2.3.4
[Make use of the differential equation
dy2 dy
which is satisfied by u — cosmx and by u = sinmx, where y = sinx.J
8. From the preceding formulae deduce developments for the functions
cos (n arc cos x) , sin (n arc cos x) .
CHAPTER X
PLANE CURVES
The curves and surfaces treated in Analytic Geometry, properly
speaking, are analytic curves and surfaces. However, the geomet
rical concepts which we are about to consider involve only the exist
ence of a certain number of successive derivatives. Thus the curve
whose equation is y = f(x) possesses a tangent if the function f(x)
has a derivative /'(#) ; it has a radius of curvature if /'(#) has a
derivative /"(cc); and so forth.
I. ENVELOPES
201. Determination of envelopes. Given a plane curve C whose
equation
(1) f(x, y, a) = 0
involves an arbitrary parameter a, the form and the position of the
curve will vary with a. If each of the positions of the curve C is
tangent to a fixed curve E, the curve E is called the envelope of the
curves C, and the curves C are said to be enveloped by E. The
problem before us is to establish the existence (or non-existence) of
an envelope for a given family of curves C, and to determine that
envelope when it does exist.
Assuming that an envelope E exists, let (x, y] be the point of tan-
gency of E with that one of the curves C which corresponds to a cer
tain value a of the parameter. The quantities x and y are unknown
functions of the parameter a, which satisfy the equation (1). In
order to determine these functions, let us express the fact that the
tangents to the two curves E and C coincide for all values of a.
Let Sx and 8y be two quantities proportional to the direction cosines
of the tangent to the curve C, and let dx/da and dy/da be the
derivatives of the unknown functions x = <£(«), y = $(&)• Then a
necessary condition for tangency is
dx dy
da _ da
C *'
426
X, §201] ENVELOPES 427
On the other hand, since a in equation (1) has a constant value for
the particular curve C considered, we shall have
(3) ¥sx + (^Sy = 0,
dx dy "
which determines the tangent to C. Again, the two unknown func
tions x = <£(«), y = ^(«) satisfy the equation
f(x, y, a) = 0,
also, where a is now the independent variable. Hence
cfdx cfdt/ df
(4) oIT- + aj+a=0»
ex da cy da oa
or, combining the equations (2), (3), and (4),
The unknown functions x = <£(«), y = \f/(d) are solutions of this equa
tion and the equation (1). Hence the equation of the envelope, in
case an envelope exists, is to be found by eliminating the parameter a
between the equations f = 0, df/da = 0.
Let R(x, y) — 0 be the equation obtained by eliminating a between
(1) and (5), and let us try to determine whether or not this equation
represents an envelope of the given curves. Let C0 be the particu
lar curve which corresponds to a value a0 of the parameter, and let
(a-0, y0) be the coordinates of the point Mti of intersection of the
two curves
(6) /(*, y, a0) = 0, £*- = <>.
t/a0
The equations (1) and (5) have, in general, solutions of the form
x = <f>(a), y = \j/(a), which reduce to x0 and y0, respectively, for
a — a0. Hence for a — aQ we shall have
dx0 \da/o c //Q \da/o
This equation taken in connection with the equation (3) shows
that the tangent to the curve C0 coincides with the tangent to the
curve described by the point (x, y*), at least unless df/dx and df/dy
are both zero, that is, unless the point MQ is a singular point for the
curve C0. It follows that the equation R(x, y) = 0 represents either
the envelope of the curves C or else the locus of singular points on
these curves,
428 PLANE CURVES [X, §202
This result may be supplemented. If each of the curves C has
one or more singular points, the locus of such points is surely a part
of the curve R(x, y) = 0. Suppose, for example, that the point (x, y)
is such a singular point. Then x and y are functions of a which
satisfy the three equations
^\ /• Q />
f(x,y,a) = Q, ^ = 0, ^ = 0,
and hence also the equation df/da = 0. It follows that x and y
satisfy the equation R(x, y) = 0 obtained by eliminating a between
the two equations / = 0 and df/Sa = 0. In the general case the
curve R(XJ y) = 0 is composed of two analytically distinct parts,
one of which is the true envelope, while the other is the locus of
the singular points.
Example. Let us consider the family of curves
/(*,*/,«) = /-2/2 + (* -«)2 = °-
The elimination of a between this equation and the derived equation
gives y* — y2 = 0, which represents the three straight lines y = 0,
y — jf. l, y — — 1. The given family of curves may be generated
by a translation of the curve y* — y2 + x2 = 0 along the x axis.
This curve has a double point at the origin, and it is tangent to
each of the straight lines y = ± I at the points where it cuts the
y axis. Hence the straight line y = 0 is the locus of double points,
whereas the two straight lines y = ± 1 constitute the real envelope.
202. If the curves C have an envelope E, any point of the envelope
is the limiting position of the point of intersection of two curves of
the family for which the values of the parameter differ by an infini
tesimal. For, let
(7) f(x, y, a) = 0, f(x,y,a + h) = 0
be the equations of two neighboring curves of the family. The
equations (7), which determine the points of intersection of the two
curves, may evidently be replaced by the equivalent system
f(x, y, a + h)-f(x,y, a) _
f(x, y, a) = 0, — - 0,
X, §202] ENVELOPES 429
the second of which reduces to Sf/da = 0 as h approaches zero, that
is, as the second of the two curves approaches the first. This prop
erty is fairly evident geometrically. In Fig. 37, a, for instance, the
point of intersection N of the two neighboring curves C and C'
approaches the point of tangency M as C' approaches the curve C
FIG. 37, b
as its limiting position. Likewise, in Fig. 37, b, where the given
curves (1) are supposed to have double points, the point of intersec
tion of two neighboring curves C and C' approaches the point where
C cuts the envelope as C' approaches C.
The remark just made explains why the locus of singular points
is found along with the envelope. For, suppose that f(x, y, a) is a
polynomial of degree m in a. For any point MQ(x0, ?/0) chosen at
random in the plane the equation
(8) /(.r0, 7,0,0 = 0
will have, in general, m distinct roots. Through such a point there
pass, in general, m different curves of the given family. But if the
point M0 lies on the curve R (or, y) = 0, the equations
are satisfied simultaneously, and the equation (8) has a double root.
The equation R(x, y) = 0 may therefore be said to represent the
locus of those points in the plane for which two of the curves of
the given family which pass through it have merged into a single
one. The figures 37, a, and 37, b, show clearly the manner in which
two of the curves through a given point merge into a single one as
that point approaches a point of the curve R(x, y) = 0, whether on
the true envelope or on a locus of double points.
430 PLANE CURVES [X, §203
Note. It often becomes necessary to find the envelope of a family
of curves
(9) F(x, y, a, ft) = 0
whose equation involves two variable parameters a and b, which
themselves satisfy a relation of the form <f>(a, ft) = 0. This case
does not differ essentially from the preceding general case, however,
for ft may be thought of as a function of a defined by the equation
<£ = 0. By the rule obtained above, we should join with the given
equation the equation obtained by equating to zero the derivative
of its left-hand member with respect to a :
dp dFdt __n
~0 I 07 7 - "•
ca co da
But from the relation <f>(a, ft) = 0 we have also
d(fr d<f> db _
da db da
whence, eliminating db/da, we obtain the equation
_
da db db da
which, together with the equations F = 0 and <f> = 0, determine the
required envelope. The parameters a and ft may be eliminated
between these three equations if desired.
203. Envelope of a straight line. As an example let us consider the equation
of a straight line D in normal form
(11) zcosa + y sin a — f(a) — 0,
where the variable parameter is the angle a. Differentiating the left-hand side
with respect to this parameter, we find as the second equation
(12) — xsinct + y cos a — f(a) = 0.
These two equations (11) and (12) determine the point of intersection of any
one of the family (11) with the envelope E in the form
(x=f(a)cosa -f '(a) sin a,
\ y = f(a) sin a + f'(a) cos a .
It is easy to show that the tangent to the envelope E which is described by this
point (z, y) is precisely the line D. For from the equations (13) we find
( dx = - [/(«) + /"(«)] sin a da ,
I dy = [/(a) + /"(a)] cos a da ,
whence dy/dx = — cot a , which is precisely the slope of the line D.
X, § 203]
ENVELOPES
431
Moreover, if s denote the length of the arc of the envelope from any fixed
point upon it, we have, from (14),
ds =
whence
+ dy* = ± [/(a) + /"(a)] da- ,
= ± [//(or) da +/'(<*)].
Hence the envelope will be a curve which is easily rectifiable if we merely choose
for /(a) the derivative of a known function.*
As an example let us set f(a) = I sin a cos a. Taking y = 0 and x = 0 suc
cessively in the equation (11), we find (Fig. 38) OA = I sin a, OB = Zcosa,
respectively ; hence AB = I. The required
curve is therefore the envelope of a straight
line of constant length Z, whose extremities
always lie on the two axes. The formulae
(13) give in this case
x = I sin3 a , y = I cos3 a- ,
and the equation of the envelope is
-A* AA?
if*®-1'
which represents a hypocycloid with four
cusps, of the form indicated in the figure.
As a varies from 0 to 7T/2, the point of con
tact describes the arc DC. Hence the length of the arc, counted from D, is
s= I 3 1 sin a cos a da = — sin2 a.
Jo 2
Let I be the fourth vertex of the rectangle determined by OA and OB, and M
the foot of the perpendicular let fall from / upon AB. Then, from the tri
angles AMI and APM, we find, successively,
AM = AI cos a = I cos2 a , AP = AM sin a = I cos2 a sin a .
Hence OP = OA — AP = I sin3 a, and the point M is the point of tangency of
the line AB with the envelope. Moreover
BM = l-AM=lsm*a;
hence the length of the arc DM = 3BM/2.
pIG
* Each of the quantities which occur in the formula for s, s = f'(a) + ff(a) daf,
has a geometrical meaning : a is the angle between the x axis and the perpendicular
ON let fall upon the variable line from the origin ; f(<x) is the distance ON from the
origin to the variable line; and /'(<*) is, except for sign, the distance MN from
the point M where the variable line touches its envelope to the foot N of the perpen
dicular let fall upon the line from the origin. The formula for 5 is often called
Legendre's formula.
432
PL AXE CURVES
[X, § 204
204. Envelope of a circle. Let us consider the family of circles
where a, 6, and p are functions of a variable parameter t. The points where a
circle of this family touches the envelope are the points of intersection of the
circle and the straight line
(16) (x - a) a' + (y - b) b' + pp' = 0 .
This straight line is perpendicular to the tangent M T to the curve C described
by the center (a, 6) of the variable circle (15), and its distance from the center is
p dp/ds, where s denotes the length of
y the arc of the curve C measured from
some fixed point. Consequently, if the
line (16) meets the circle in the two
points N and N', the chord NN' is
/<j[ bisected by the tangent M T at right
/ angles. It follows that the envelope
'' consists of two parts, which are, in
general, branches of the same analytic
curve. Let us now consider several
special cases.
1) If p is constant, the chord of con
tact NN' reduces to the normal PP' to
the curve C, and the envelope is com
posed of the two parallel curves C\ and
C{ which are obtained by laying off the constant distance p along the normal,
on either side of the curve C.
2) If p = s + K, we have p dp/ds = p, and the chord NN' reduces to the tan
gent to the circle at the point Q. The two portions of the envelope are merged
into a single curve T, whose normals are tangents to the curve C. The curve C
is called the evolute of T, and, conversely, T is called an involute of C (see § 206),
If dp > ds, the straight line (16) no longer cuts the circle, and the envelope is
imaginary.
Secondary caustics. Let us suppose that
the radius of the variable circle is propor
tional to the distance from the center to a
fixed point 0. Taking the fixed point O as
the origin of coordinates, the equation of the
circle becomes
where A; is a constant factor, and the equation
of the chord of contact is
FIG. 39
FIG. 40
E
(x - a) a' + (y - b) b' + k*(aa' + bb') = 0.
If 8 and 8' denote the distances from the
center of the circle to the chord of contact and to the parallel to it through the
origin, respectively, the preceding equation shows that 5 = k2d'. Let P be a
point on the radius MO (Fig. 40), such that MP = k*MO, and let C be the
X,§'-'05]
CURVATURE 433
locus of the center. Then the equation just found shows that the chord of con
tact is the perpendicular let fall from P upon the tangent to C at the center M.
Let us suppose that k is less than unity, and let E denote that branch of the
envelope which lies on the same side of the tangent MT as does the point 0.
Let i and r, respectively, denote the two angles which the two lines MO and
MN make with the normal MI to the curve C. Then we shall have
. _ M q _ Mp sin i _ Mq _ MQ _ 1
= ' ~' ~~ M~ MP~ k
Now let us imagine that the point 0 is a source of light, and that the curve C
separates a certain homogeneous medium in which O lies from another medium
whose index of refraction with respect to the first is l/k. After refraction the
incident ray OM will be turned into a refracted ray MR, which, by the law of
refraction, is the extension of the line NM. Hence all the refracted rays MR
are normal to the envelope, which is called the secondary caustic of refraction.
The true caustic, that is, the envelope of the refracted rays, is the evolute of the
secondary caustic.
The second branch E' of the envelope evidently has no physical meaning ;
it would correspond to a negative index of refraction. If we set k — 1, the
envelope E reduces to the single point 0, while the portion E' becomes the locus
of the points situated symmetrically with 0 with respect to the tangents to C.
This portion of the envelope is also the secondary caustic of reflection for inci
dent rays reflected from C which issue from the fixed point 0. It may be shown
in a manner similar to the above that if a circle be described about each point of
C with a radius proportional to the distance from its center to a fixed straight
line, the envelope of the family will be a secondary caustic with respect to a
system of parallel rays.
II. CURVATURE
205. Radius of curvature. The first idea of curvature is that the
curvature of one curve is greater than that of another if it recedes
more rapidly from its tangent. In order to render this somewhat
vague idea precise, let us first consider the case of a circle. Its
curvature increases as its radius diminishes ; it is therefore quite
natural to select as the measure of its curvature the simplest func
tion of the radius which increases as the radius diminishes, that
is, the reciprocal l/R of the radius. Let AB be an arc of a circle
of radius R which subtends an angle «o at the center. The angle
between the tangents at the extremities of the arc AB is also <o, and
the length of the arc is s = R<a. Hence the measure of the curva
ture of the circle is w/s. This last definition may be extended to
an arc of any curve. Let AB be an arc of a plane curve without a
point of inflection, and w the angle between the tangents at the
extremities of the arc, the directions of the tangents being taken
in the same sense according to some rule, — the direction from A
434
PLANE CURVES
[X,§205
FIG. 41
toward B, for instance. Then the quotient w/arc AB is called the
average curvature of the arc AB. As the point B approaches the
point A this quotient in general approaches a limit, which is called
the curvature at the point A. The
radius of curvature at the point A is
defined to be the radius of the circle
which would have the same curvature
which the given curve has at the point
A ; it is therefore equal to the recipro
cal of the curvature. Let s be the
— length of the arc of the given curve
measured from some fixed point, and
a the angle between the tangent and
some fixed direction, — the x axis, for example. Then it is clear
that the average curvature of the arc AB is equal to the absolute
value of the quotient Aa/A.s ; hence the radius of curvature is given
by the formula
, ,. As ds
R = ± lim — = ± — -•
Aa art
Let us suppose the equation of the given curve to be solved for y
in the form y =f(x). Then we shall have
y"dx
a = arc tan y', da = y _ " .„» ds =
and hence
(17> *=± y,,
Since the radius of curvature is essentially positive, the sign ±
indicates that we are to take the absolute value of the expression
on the right. If a length equal to the radius of curvature be laid
off from A upon the normal to the given curve on the side toward
which the curve is concave, the extremity / is called the center of
curvature. The circle described about 7 as center with R as radius
is called the circle of curvature. The coordinates (x0, y0) of the
center of curvature satisfy the two equations
which express the fact that the point lies on the normal at a dis
tance R from A. From these equations we find, on eliminating xl}
X, § 205]
CURVATURE 435
In order to tell which sign should be taken, let us note that if y" is
positive, as in Fig. 41, y^ — y must be positive ; hence the positive
sign should be taken in this case. If y" is negative, y± — y is nega
tive, and the positive sign should be taken in this case also. The
coordinates of the center of curvature are therefore given by the
formulae
1 + y'2 , 1 + y'2
(18) yi-9*=—jf-> **-*=- y'—fT-
When the coordinates of a point (x, y) of the variable curve are
given as functions of a variable parameter t, we have, by § 33,
and the formulae (17) and (18) become
jz_± (dx* + dy*?
. j ** y
} ' dy(dx* + dif~)
&x i dxd2y-dyd*x
At a point of inflection y" = 0, and the radius of curvature is
infinite. At a cusp of the first kind y can be developed according
to powers of xl/2 in a series which begins with a term in x ; hence
y' has a finite value, but y" is infinite, and therefore the radius of
curvature is zero.
Note. When the coordinates are expressed as functions of the arc s of the
curve,
x = <P(s) , y = f (a) ,
the functions <f> and f satisfy the relation
0'2(S) + V/2(S)=1,
since dx2 + dy2 = da2, and hence they also satisfy the relation
<£'(/>" + -fy'^i" = 0 .
Solving these equations for <£' and \f/', we find
where e = ± 1, and the formula for the radius of curvature takes on the espe
cially elegant form
(20) s»[*
436 PLANE CURVES [X, §206
206. Evolutes. The center of curvature at any point is the limit
ing position of the point of intersection of the normal at that point
with a second normal which approaches the first one as its limiting
position. For the equation of the normal is
where X and Y are the running coordinates. In order to find the
limiting position of the point of intersection of this normal with
another which approaches it, we must solve this equation simulta
neously with the equation obtained by equating the derivative of the
left-hand side with respect to the variable parameter x, i.e.
The value of Y found from this equation is precisely the ordinate
of the center of curvature, which proves the proposition. It follows
that the locus of the center of curvature is the envelope of the
normals of the given curve, i.e. its evolute.
Before entering upon a more precise discussion of the relations
between a given curve and its evolute, we shall explain certain con
ventions. Counting the length of the arc of the given curve in a
definite sense from a fixed point as origin, and denoting by a the
angle between the positive direction of the x axis and the direction
of the tangent which corresponds to increasing values of the arc,
we shall have tan a = ± y', and therefore
cos a
On the right the sign -f- should be taken, for if x and s increase
simultaneously, the angle a is acute, whereas if one of the varia
bles x and s increases as the other decreases, the angle « is obtuse
(§ 81). Likewise, if (3 denote the angle between the y axis and the
tangent, cos (3 = dy/ds. The two formulae may then be written
dx dy
COS or = — > Sin a = ~ -j
as as
where the angle a is counted as in Trigonometry.
On the other hand, if there be no point of inflection upon the
given arc, the positive sense on the curve may be chosen in such a
way that s and a increase simultaneously, in which case R = ds/da
all along the arc. Then it is easily seen by examining the two
possible cases in an actual figure that the direction of the segment
X, § 206]
CURVATURE
437
starting at the point of the curve and going to the center of curva
ture makes an angle a1 = a -f 7r/2 with the x axis. The coordinates
(XL > y\) °f the center of curvature are therefore given by the formulae
xl = x + R cos ( a + — } = x — R sin a,
\ •"/
(\
a + —\ = y + R cos a,
whence we find
cfaj = cos ads — R cos a da — sin a dR = — sin a c£R,
efo/! = sin ads — R sin a tZa + cos a dR = cos a rf7? .
In the first place, these formulas show that dyl/dxl = — cot a, which
proves that the tangent to the evolute is the normal to the given
curve, as we have already seen. Moreover
ds\ = dx\ + dy\ = dR* ,
or dsi = ± dR. Let us suppose for definiteness that the radius
of curvature constantly increases as we proceed along the curve C
(Fig. 42) from Ml to Jl/2, and let us choose the positive sense of
motion upon the evolute (D) in such a way
that the arc s1 of (Z)) increases simultane
ously with R. Then the preceding formula
becomes d$i = dR, whence st = R + C. It
follows that the arc /!/2 = R2 — R1} and we
see that the length of any arc of the evolute
is equal to the difference between the tivo
radii of curvature of the curve C which cor
respond to the extremities of that arc.
This property enables us to construct the
involute C mechanically if the evolute (Z>) be
given. If a string be attached to (D) at an
arbitrary point A and rolled around (D) to /2, thence following the
tangent to Mt, the point M2 will describe the involute C as the
string, now held taut, is wound further on round (Z>). This con
struction may be stated as follows : On each of the tangents IM of
the evolute lay off a distance IM = I, where I + s = const., s being
the length of the arc AI of the evolute. Assigning various values
to the constant in question, an infinite number of involutes may be
drawn, all of which are obtainable from any one of them by laying
off constant lengths along the normals.
M
FIG. 42
438
PLANE CURVES
[X, §207
All of these properties may be deduced from the general formula
for the differential of the length of a straight line segment (§ 82)
dl ••= — dar1 COS (D! — (Zcr2 COS o>.2 .
If the segment is tangent to the curve described by one of its
extremities and normal to that described by the other, we may set
<0l — TT, w.2 = 7T/2, and the formula becomes dl — da^ = 0. If the
straight line is normal to one of the two curves and Z is constant,
dl = 0, cos <DI = 0, and therefore cos o>2 = 0.
The theorem stated above regarding the arc of the evolute depends
essentially upon the assumption that the radius of curvature con
stantly increases (or decreases) along the whole arc considered. If
this condition is not satisfied, the statement of the theorem must
be altered. In the first place, if the radius of curvature is a maxi
mum or a minimum at any point, dR = 0 at that point, and hence
dxi = dyl = 0. Such a point is a cusp on
the evolute. If, for example, the radius
of curvature is a maximum at the point M
(Fig. 43), we shall have
arc//! = IM — A-!/!,
arc 77, = IM - 72 3/a ,
whence
arc /! 7/2 = 2 737 —
- 72
Hence the difference 7t M± — 72 M2 is equal
to the difference between the two arcs Hi and 772 and not their sum.
207. Cycloid. The cycloid is the path of a point upon the circumference of a
circle which rolls without slipping on a fixed straight line. Let us take the
n'
Fio. 44
fixed line as the x axis and locate the origin at a point where the point chosen on
the circle lies in the x axis. When the circle has rolled to the point 7 (Fig. 44)
the point on the circumference which was at 0 has come into the position M,
X, §207] CURVATURE 439
where the cirenlar arc IM is equal to the segment OI. Let us take the angle 0
between the radii CM and CI as the variable parameter. Then the coordinates
of the point M are
x = 01 - IP = R(f> - E sin 0, y - MP = 1C + CQ = R - R cos 0,
where P and Q are the projections of M on the two lines 01 and IT, respec
tively. It is easy to show that these formulae hold for any value of the angle 0.
In one complete revolution the point whose path is sought describes the arc
OBO\. If the motion be continued indefinitely, we obtain an infinite number
of arcs congruent to this one. From the preceding formulae we find
x = R(<}> — sin <p) , dx = R(l — cos <(>) d<f> , d-x = Rsin0d02,
y = R(l — cos 0) , dy = R sin 0 d0 , d*y = .Rcos0 d02,
and the slope of the tangent is seen to be
dy sin ^ 0
• — ~ • = cot — i
dx 1 — cos 0 2
which shows that the tangent at M is the straight line M T, since the angle
MTC = 0/2, the triangle MTC being isosceles. Hence the normal at M is the
straight line MI through the point of tangency I of the fixed straight line with
the moving circle. For the length of the arc of the cycloid we find
(Zs2 = E2 d02 [sin20 + (1 - cos 0)2] = 4R2 sin2 - d02 or ds = 2R sin ~ d0 ,
2 2i
if the arc be counted in the sense in which it increases with 0. Hence, counting
the arc from the point 0 as origin, we shall have
= 4fi( 1-cos-Y
V 2/
Setting <f> = 2it, we find that the length of one whole section OBOt is 8R. The
length of the arc OMB from the origin to the maximum B is therefore 4.R, and
the length of the arc BM (Fig. 44) is 4R cos 0/2. From the triangle MTC the
length of the segment MT is 2R cos 0/2 ; hence arc BM — 2MT.
Again, the area up to the ordinate through M is
A= f ydx= C E2(l-2cos0 + cos20)d0
Jo Jo
/3 sin 20\ „.
A= -0-2sm0 + - ^)#2.
V" 4 /
Hence the area bounded by the whole arc OBOi and the base OOi is 37T.K2, that
is, three times the area of the generating circle. (GALILEO.)
The formula for the radius of curvature of a plane curve gives for the cycloid
440
PLANE CURVES
[X, § 208
On the other hand, from the triangle MCI, MI = 2R sin 0/2. Hence p = 23fl,
and the center of curvature may be found by extending the straight line MI
past I by its own length. This fact enables us to determine the evolute easily.
For, consider the circle which is symmetrical to the generating circle with
respect to the point I. Then the point M' where the line MI cuts this second
circle is evidently the center of curvature, since M'l = MI. But we have
arc T'M' = TtR - arc IM' = TtR - arc IM = nE - 01,
or
arc T'M' = OH - OI = IH = T'E' .
Hence the point M' describes a cycloid which is congruent to the first one, the
cusp being at B' and the maximum at O. As the point M describes the arc
BOi, the point M' describes a second arc B'Oi which is symmetrical to the arc
OB' already described, with respect to BB'.
208. Catenary. The catenary is the plane curve whose equation with respect
to a suitably chosen set of rectangular axes is
Its appearance is indicated by the arc MAM' in the figure (Fig. 45).
From (21) we find
FIG. 45
4 a2
If 0 denote the angle which the tangent TM makes with the x axis, the formula
for y' gives
X X
_ ea — e~ a 2 a
— - , cos0 = — — = -'
ea + e a en + e "
The radius of curvature is given by the formula
y" a
But, in the triangle MPN, MP = MNcostf,- hence
COS 0
X, §209] CURVATURE 441
It follows that the radius of curvature of the catenary is equal to the length of
the normal MN. The evolute may be found without difficulty from this fact.
The length of the arc AM of the catenary is given by the formula
px x _y
/ ea + e « , a I * _ *\
dx = - ( ea — e « 1 .
Jo 2 2\ /
or s = y sin <f>. If a perpendicular Pm be dropped from P (Fig. 46) upon the
tangent MT, we find, from the triangle PMm,
Mm = MP sin <f> = s .
Hence the arc AM is equal to the distance Mm.
209. Tractrix. The curve described by the point m (Fig. 45) is called the
tractrix. It is an involute of the catenary and has a cusp at the point A. The
length of the tangent to the tractrix is the distance mP. But, in the triangle
MPm, mP = y cos</> = a ; hence the length mP measured along the tangent to
the tractrix from the point of tangency to the x axis is constant and equal to a.
The tractrix is the only curve which has this property.
Moreover, in the triangle M TP, Mm x mT = a2. Hence the product of the
radius of curvature and the normal is a constant for the tractrix. This property
is shared, however, by an infinite number of other plane curves.
The coordinates (xi, y\) of the point TO are given by the formulae
eu _ e~a
x\ = x — s cos (f> = x — a •
or, setting ex/a = tan 0/2, the equations of the tractrix are
(22) xi — a cos 6 + a log (tan-
As the parameter 8 varies from n/2 to n, the point (xl , yv) describes the arc
Amn, approaching the x axis as asymptote. As 0 varies from n/2 to 0, the
point (xi , yi) describes the arc Am'n', symmetrical to the first with respect to
the y axis. The arcs Amn and Am'n' correspond, respectively, to the arcs AM
and AM' of the catenary.
210. Intrinsic equation. Let us try to determine the equation of a plane
curve when the radius of curvature R is given as a function of the arc s,
E = <f>(s). Let a be the angle between the tangent and the x axis ; then
R = ± ds/da, and therefore
. ds . ds
da = ± — =±
A first integration gives
/'
«/»„
ds
a = tt0 ± | — - '.
*(«)
442 PLANE CURVES [X, §210
and two further integrations give x and y in the form
x — XQ + I cos a ds , y — y0 + f sin a ds .
J*o J*o
The curves defined by these equations depend upon the three arbitrary con
stants x0, 2/o, and a0. But if we disregard the position of the curve and think
only of its form, we have in reality merely a single curve. For, if we first con
sider the curve C defined by the equations
the general formulae may be written in the form
x = x0 + X cos O-Q —
y = 2/0 + X sin <*o+ Y cos cr0»
if the positive sign be taken. These last formulae define simply a transforma
tion to a new set of axes. If the negative sign be selected, the curve obtained
is symmetrical to the curve C with respect to the X axis. A plane curve is
therefore completely determined, in so far as its form is concerned, if its radius
of curvature be known as a function of the arc. The equation B — #(s) is
called the intrinsic equation of the curve. More generally, if a relation between
any two of the quantities B, s, and a be given, the curve is completely deter
mined in form, and the expressions for the coordinates of any point upon it
may be obtained by simple quadratures.
For example, if B be known as a function of a, B =/(«), we first find
ds = /(a) da, and then
dx = cos a f (a) da,
dy = sin af(a) da ,
whence x and y may be found by quadratures. If J? is a constant, for instance,
these formulae give
x = XQ + B sin a , y = y0 — B cos a ,
and the required curve is a circle of radius B. This result is otherwise evident
from the consideration of the evolute of the required curve, which must reduce
to a single point, since the length of its arc is identically zero.
As another example let us try to find a plane curve whose radius of curva
ture is proportional to the reciprocal of the arc, B = a?/s. The formulas give
and then
r s s ds _ i
=I * *
Although these integrals cannot be evaluated in explicit form, it is easy to gain
an idea of the appearance of the curve. As s increases from 0 to + cc, x and y
each pass through an infinite number of maxima and minima, and they approach
the same finite limit. Hence the curve has a spiral form and approaches
asymptotically a certain point on the line y = x.
X,§2ll] CONTACT OSCULATION 443
III. CONTACT OF PLANE CURVES
211. Order of contact. Let C and C' be two plane curves which
are tangent at some point A. To every point m on C let us assign,
according to any arbitrary law whatever, a point m' on C', the only
requirement being that the point m'
should approach A with m. Taking
the a.TcAm — or, what amounts to
the same thing, the chord Am — as
the principal infinitesimal, let us first
investigate what law of correspond
ence will make the order of the infin
itesimal mm' with respect to Am as
large as possible. Let the two curves p
be referred to a system of rectangular
or oblique cartesian coordinates, the axis of y not being parallel to the
common tangent AT. Let
(C) </=/(*),
(C") Y=F(x)
be the equations of the two curves, respectively, and let (o»0, y0) be
the coordinates of the point A. Then the coordinates of m will
be [a-0 -f h, f(x0 -f A)], and those of m' will be [#0 + k, F(xQ + &)],
where k is a function of h which defines the correspondence between
the two curves and which approaches zero with h.
The principal infinitesimal Am may be replaced by h = ap, for
the ratio ap/Am approaches a finite limit different from zero as the
point m approaches the point A. Let us now suppose that mm' is
an infinitesimal of order r -f 1 with respect to h, for a certain
method of correspondence. Then mm' is of order 2r + 2. If 6
denote the angle between the axes, we shall have
mm1 = [F(x0 + jfe) — /(*„ + *) + (k- h) cos OJ + (k — A)2 sin2 6 ;
hence each of the differences k — h and F(x0 + k) —f(x0 + A) must
be an infinitesimal of order not less than r -f 1, that is,
k = h -f ah,* + l, F(x0 + k) -f(x0 + A) = ph'+\
where a and /3 are functions of h which approach finite limits as A
approaches zero. The second of these formulae may be written in
the form
F(x0 + h + a/t^1) -f(x0 + A) = fih'*1.
PLANE CURVES [X,§2ii
If the expression F(x0 + h + ahr+l) be developed in powers of a,
the terms which contain a form an infinitesimal of order not less
than r -f 1. Hence the difference
& = F(xQ + h)-f(x0 + h)
is an infinitesimal whose order is not less than r + 1 and may exceed
r + 1. But this difference A is equal to the distance mn between
the two points in which the curves C and C" are cut by a parallel
to the y axis through ra. Since the order of the infinitesimal mm'
is increased or else unaltered by replacing m' by n, it follows that
the distance between two corresponding points on the two curves is an
infinitesimal of the greatest possible order if the two corresponding
points always lie on a parallel to the y axis. If this greatest possi
ble order is r + 1, the two curves are said to have contact of order r
at the point A.
Notes. This definition gives rise to several remarks. The y axis
was any line whatever not parallel to the tangent A T. Hence, in
order to find the order of contact, corresponding points on the two
curves may be defined to be those in which the curves are cut by
lines parallel to any fixed line D which is not parallel to the tan
gent at their common point. The preceding argument shows that
the order of the infinitesimal obtained is independent of the direc
tion of 7), — a conclusion which is easily verified. Let mn and mm'
be any two lines through a point m of the curve C which are not
parallel to the common tangent (Fig. 46). Then, from the triangle
mm' n,
mm' sin
mn sin mm' n
As the point m approaches the point A, the angles mnm' and mm'n
approach limits neither of which is zero or TT, since the chord m'n
approaches the tangent AT. Hence mm'/mn approaches a finite
limit different from zero, and mm' is an infinitesimal of the same
order as mn. The same reasoning shows that mm' cannot be of
higher order than mn, no matter what construction of this kind is
used to determine m' from m, for the numerator sin mnm' always
approaches a finite limit different from zero.
The principal infinitesimal used above was the arc Am or the
chord Am. We should obtain the same results by taking the arc
An of the curve C' for the principal infinitesimal, since Am and An
are infinitesimals of the same order.
X, §212] CONTACT OSCULATION 445
If two curves C and C' have a contact of order r, the points m'
on C' may be assigned to the points m on C in an infinite number
of ways which will make mm1 an infinitesimal of order r + 1, — for
that purpose it is sufficient to set k = h + aha + l, where s^r and
where a is a function of h which remains finite for h = 0. On the
other hand, if s < r, the order of mm' cannot exceed 5 -f 1.
212. Analytic method. It follows from the preceding section that
the order of contact of two curves C and C' is given by evaluating
the order of the infinitesimal
with respect to h. Since the two curves are tangent at A,
F(x^) = /(£„) and F'(x0) =f'(x0). It may happen that others of the
derivatives are equal at the same point, and we shall suppose for
the sake of generality that this is true of the first n derivatives :
(23} \ F(x0)=f(xo), F'(*0) =/'(*„),
I F"(*0) =/"(*„), • -., F^(x0) =/«>(x0),
but that the next derivatives -F(n + 1) (cc0) and /(n + 1)(a-0) are unequal.
Applying Taylor's series to each of the functions F(x) and f(x), we
find
y =
or, subtracting,
(24) F ~ y = 1.2.^(
where c and e' are infinitesimals. It follows that the order of contact
of two curves is equal to the order n of the highest derivatives of F(x)
and f(x) which are equal for x = x0.
The conditions (23), which are due to Lagrange, are the necessary
and sufficient conditions that x — x0 should be a multiple root of
order n + 1 of the equation F(z) =/(<r). But the roots of this
equation are the abscissae of the points of intersection of the two
446 PLANE CURVES [X, § 212
curves C and C" ; hence it may be said that two curves which have
contact of order n have n + 1 coincident points of intersection.
The equation (24) shows that F — y changes sign with h if n is
even, and that it does not if n is odd. Hence curves which have
contact of odd order do not cross, but curves which have contact of
even order do cross at their point of tangency. It is easy to see why
this should be true. Let us consider for definiteness a curve C'
which cuts another curve C in three points near the point A. If
the curve C" be deformed continuously in such a way that each of
the three points of intersection approaches A, the limiting position
of C' has contact of the second order with (7, and a figure shows that
the two curves cross at the point A. This argument is evidently
general.
If the equations of the two curves are not solved with respect to
Y and y, which is the case in general, the ordinary rules for the
calculation of the derivatives in question enable us to write down
the necessary conditions that the curves should have contact of
order n. The problem is therefore free from any particular diffi
culties. We shall examine only a few special cases which arise
frequently. First let us suppose that the equations of each of the
curves are given in terms of an auxiliary variable
(X = f(u),
(c) " A
and that ^(£0) = <K^o) an(^ 'A'(^o) = <£'(Xo)> i-e- that the curves are tan
gent at a point A whose coordinates are f(t0), <f>(t(,)- Iff'(to) w not
zero, as we shall suppose, the common tangent is not parallel to the
y axis, and we may obtain the points of the two curves which have
the same abscissae by setting u = t. On the other hand, x — x0 is of
the first order with respect to t — t0, and we are led to evaluate the
order of i/f(£) — <£(£) with respect to t — t0. In order that the two
curves have at least contact of order n, it is necessary and sufficient
that we should have
(25) ,K*0) = <X*o) , *'(<o) = *'(<b) , • • • , *°° (*o) - *°° Co) ,
and the order of contact will not exceed n if the next derivatives
<A(n + 1)(*o) and 4>(n + 1)(*o) are unequal.
Again, consider the case where the curve C is represented by the
two equations
(26) *=/(0, y =
X, § 212]
CONTACT OSCULATION
447
and the curve C' by the single equation F(x, y} = 0. This case may
be reduced to the preceding by replacing x in F(x, y) by /(£) and
considering the implicit function y — ^(t) defined by the equation
(27) F[/(0, «KO] = o.
Then the curve C1 is also represented by two equations of the form
(28) *=/(*)> y = ^CO-
In order that the curves C and C' should have contact of order n at
a point A which corresponds to a value tn of the parameter, it is
necessary that the conditions (25) should be satisfied. But the
successive derivatives of the implicit function i/r(£) are given by the
equations
(29)
[/'«?+ 2
Hence necessary conditions for contact of order n will be obtained
by inserting in these equations the relations
t = t0, x ==y(£0), i/^o) == <£(^o
The resulting conditions may be expressed as follows :
Let
e ^wo given curves will have at least contact of order n if and
only if
(30) F(<o) = 0, F(*o) = 0, ..-, R")(^) = 0.
The roots of the equation F(£) = 0 are the values of t which cor
respond to points of intersection of the two given curves. Hence
the preceding conditions amount to saying that t = t0 is a multiple
root of order n, i.e. that the two curves have n + 1 coincident points
of intersection.
PL AXE CURVES [X,§213
213. Osculating curves. Given a fixed curve C and another curve
C' which depends upon n -f- 1 parameters a, b, c, • • •, I,
(31) F(x, y, a, b, c, •••, t) = 0,
it is possible in general to choose these n + 1 parameters in such a
way that C' and C shall have contact of order n at any preassigned
point of C. For, let C be given by the equations x =/(£), y = <KO-
Then the conditions that the curves C and C' should have contact
of order n at the point where t = tQ are given by the equations (30),
where
F(0 =
If t0 be given, these n +1 equations determine in general the n +1
parameters a, b, c, ••-, /. The curve C' obtained in this way is
called an osculating curve to the curve C.
Let us apply this theory to the simpler classes of curves. The
equation of a straight line y = ax -+• b depends upon the two param
eters a and b ; the corresponding osculating straight lines will have
contact of the first order. If y =f(x) is the equation of the curve C,
the parameters a and b must satisfy the two equations
Axo) = ax0 + b, f'(x0) = a ;
hence the osculating line is the ordinary tangent, as we should
expect.
The equation of a circle
(32) (x - a)2 + (y - by - R* = 0
depends upon the three parameters a, b, and R ; hence the corre
sponding osculating circles will have contact of the second order.
Let y = f(x) be the equation of the given curve C ; we shall obtain
the correct values of a, b, and R by requiring that the circle should
meet this curve in three coincident points. This gives, besides the
equation (32), the two equations
(33) x-a + (y- b}y' = 0, 1+ y'2 + (y - V)y" = 0.
The values of a and b found from the equations (33) are precisely
the coordinates of the center of curvature (§ 205) ; hence the oscu
lating circle coincides with the circle of curvature. Since the con
tact is in general of order two, we may conclude that in general the
circle of curvature of a plane curve crosses the curve at their point
of tangency.
X,§2i:<] CONTACT OSCULATION 449
All the above results might have been foreseen a priori. For,
since the coordinates of the center of curvature depend only on
x, y, y\ and y", any two curves which have contact of the second
order have the same center of curvature. But the center of curva
ture of the osculating circle is evidently the center of that circle
itself; hence the circle of curvature must coincide with the oscu
lating circle. On the other hand, let us consider two circles of
curvature near each other. The difference between their radii,
which is equal to the arc of the evolute between the two centers,
is greater than the distance between the centers ; hence one of
the two circles must lie wholly inside the other, which could not
happen if both of them lay wholly on one side of the curve C in
the neighborhood of the point of contact. It follows that they
cross the curve C.
There are, however, on any plane curve, in general, certain points
at which the osculating circle does not cross the curve ; this excep
tion to the rule is, in fact, typical. Given a curve C' which depends
upon n + 1 parameters, we may add to the n + 1 equations (30) the
new equation
provided that we regard t0 as one of the unknown quantities and
determine it at the same time that we determine the parameters
a, b, c, • • • , I. It follows that there are, in general, on any plane
curve C, a certain number of points at which the order of con
tact with the osculating curve C1 is n -f 1. For example, there are
usually points at which the tangent has contact of the second order ;
these are the points of inflection, for which y" = 0. In order to find
the points at which the osculating circle has contact of the third
order, the last of equations (33) must be differentiated again, which
gives
or finally, eliminating y — b,
(34) (i+yv-*yY"-o.
The points which satisfy this last condition are those for which
dR/dx = 0, i.e. those at which the radius of curvature is a maxi
mum or a minimum. On the ellipse, for example, these points are
the vertices ; on the cycloid they are the points at which the tan
gent is parallel to the base.
450 PLANE CURVES [X,§214
214. Osculating curves as limiting curves. It is evident that an
osculating curve may be thought of as the limiting position of a
curve C" which meets the fixed curve C in n + 1 points near a fixed
point A of C, which is the limiting position of each of the points
of intersection. Let us consider for definiteness a family of
curves which depends upon three parameters a, b, and c, and let
tn + Aj , t0 + hi, and t0 + 7t3 be three values of t near t0. The curve
C' which meets the curve C in the three corresponding points is
given by the three equations
(35) F(t0 + AO = 0 , F(t0 + A2) = 0 , F(t0 + A,) = 0 .
Subtracting the first of these equations from each of the others and
applying the law of the mean to each of the differences obtained,
we find the equivalent system
(36) F(^0 + A1) = 0, F'(f0 + *i) = 0, F'(*0+fc2) = 0,
where &t lies between hl and 7/,2, and &2 between Ax and hz. Again,
subtracting the second of these equations from the third and apply
ing the law of the mean, we find a third system equivalent to either
of the preceding,
(37) Fft + AO-O, F'(f0 + *i) = 0, F"ft, + /,) = 0,
where ^ lies between 7^ and £2. As 7?u 7?2, and hs all approach
zero, &!, k2, and ^ also all approach zero, and the preceding equa
tions become, in the limit,
which are the very equations which determine the osculating curve.
The same argument applies for any number of parameters whatever.
Indeed, we might define the osculating curve to be the limiting
position of a curve C' which is tangent to C at p points and cuts C
at q other points, where 2p + q = n + 1, as all these p + q points
approach coincidence.
For instance, the osculating circle is the limiting position of a
circle which cuts the given curve C in three neighboring points. It
is also the limiting position of a circle which is tangent to C and
which cuts C at another point whose distance from the point of
tangency is infinitesimal. Let us consider for a moment the latter
property, which is easily verified.
Let us take the given point on C as the origin, the tangent at
that point as the x axis, and the direction of the normal toward the
X, Exs.]
EXERCISES
451
center of curvature as the positive direction of the y axis. At the
oiigin, y' = 0. Hence R = 1/y", and therefore, by Taylor's series,
1
where e approaches zero with x. It fol
lows that R is the limit of the expres
sion «2/(2y) = OP2/(2MP) as the Point
M approaches the origin. On the other
hand, let Rl be the radius of the circle
Ci which is tangent to the x axis at the
origin and which passes through M.
Then we shall have
Fia
or
OP = Mm = MP(2Rl - MP) ,
;2
OP
2MP
— -tti — ^ '
M_P.
2
hence the limit of the radius 7£a is really equal to the radius of
curvature R
EXERCISES
1. Apply the general formulae to find the evolute of an ellipse ; of an hyper
bola ; of a parabola.
2. Show that the radius of curvature of a conic is proportional to the cube
of the segment of the normal between its points of intersection with the curve
and with an axis of symmetry.
3. Show that the radius of curvature of the parabola is equal to twice the
segment of the normal between the curve and the directrix.
4. Let F and F' be the foci of an ellipse, M a point on the ellipse, MN the
normal at that point, and N the point of intersection of that normal and the
major axis of the ellipse. Erect a perpendicular NK to MN at jV, meeting MF
at K. At K erect a perpendicular KO to MF, meeting MN at 0. Show that
0 is the center of curvature of the ellipse at the point M.
5. For the extremities of the major axis the preceding construction becomes
illusory. Let .4 CM/ be the major axis and BO'B the minor axis of the ellipse.
On the segments OA and OB construct the rectangle OA EB. From E let fall
a perpendicular on AB, meeting the major and minor axes at C and D, respec
tively. Show that C and D are the centers of curvature of the ellipse for the
points A and B, respectively.
6. Show that the evolute of the spiral p = aemu> is a spiral congruent to the
given spiral.
452 PLANE CURVES [X, Exs.
7. The path of any point on the circumference of a circle which rolls with
out slipping along another (fixed) circle is called an epicycloid or an hypocycloid.
Show that the evolute of any such curve is another curve of the same kind.
8. Let AB be an arc of a curve upon which there are no singular points and
no points of inflection. At each point m of this arc lay off from the point m
along the normal at m a given constant length I in each direction. Let wtj and
m2 be the extremities of these segments. As the point m describes the arc AB,
the points mi and m2 will describe two corresponding arcs AiBi and AtB2.
Derive the formulae Si = S — 18, S2 = S + W, where S, <S'i , and <SY2 are the
lengths of the arcs AB, A\B\, and A2B2, respectively, and where 0 is the angle
between the normals at the points A and B. It is supposed that the arc -Ai-Bi
lies on the same side of AB as the evolute, and that it does not meet the evolute.
[Licence, Paris, July, 1879.]
9. Determine a curve such that the radius of curvatures p at any point M
and the length of the arc s — AM measured from any fixed point A on the curve
satisfy the equation as = p2 + a2, where a is a given constant length.
[Licence, Paris, July, 1883.]
10. Let C be a given curve of the third degree which has a double point
at 0. A right angle MON revolves about the point O, meeting the curve C in
two variable points M and N. Determine the envelope of the straight line MN.
In particular, solve the problem for each of the curves Xy2 = x3 and z3 + ys = f*xy.
[Licence, Bordeaux, July, 1885.]
11. Find the points at which the curve represented by the equations
x — a (nw — sin u), y = a (n — cos w)
has contact of higher order than the second with the osculating circle.
[Licence, Grenoble, July, 1885.]
12. Let m, mi , and m2 be three neighboring points on a plane curve. Find
the limit approached by the radius of the circle circumscribed about the triangle
formed by the tangents at these three points as the points approach coincidence.
13. If the evolute of a plane curve without points of inflection is a closed
curve, the total length of the evolute is equal to twice the difference between the
sum of the maximum radii of curvature and the sum of the minimum radii of
curvature of the given curve.
14. At each point of a curve lay off a constant segment at a constant angle
with the normal. Show that the locus of the extremity of this segment is a
curve whose normal passes through the center of curvature of the given curve.
15. Let r be the length of the radius vector from a fixed pole to any point of
a plane curve, and p the perpendicular distance from the pole to the tangent.
Derive the formula R = ±rdr/dp, where R is the radius of curvature.
16. Show that the locus of the foci of the parabolas which have contact of
the second order with a given curve at a fixed point is a circle.
17. Find the locus of the centers of the ellipses whose axes have a fixed direc
tion, and which have contact of the second order at a fixed point with a given
curve.
CHAPTER XI
SKEW CURVES
I. OSCULATING PLANE
215. Definition and equation. Let M T be the tangent at a point M
of a given skew curve F. A plane through MT and a point M' of
F near M in general approaches a limiting position as the point M'
approaches the point M. If it does, the limiting position of the
plane is called the osculatiny plane to the curve F at the point M.
We shall proceed to find its equation.
Let
(i) *=A9» y = *(0» *-rtO
be the equations of the curve F in terms of a parameter t, and let t
and £ + h be the values of t which correspond to the points M and
&f, respectively. Then the equation of the plane MTM1 is
A(X -*)+ B(Y-y) + C(Z- 3) = 0,
where the coefficients ^4, B, and C must satisfy the two relations
(2)
(3)
Expanding f(t + h), <j>(t + A) and \j/(t -f A) by Taylor's series, the
equation (3) becomes
After multiplying by 7^, let us subtract from this equation the equa
tion (2), and then divide both sides of the resulting equation by
A2/2. Doing so, we find a system equivalent to (2) and (3) :
<) + «a] + ^[^"(0 + cs] = 0, .
where d, e2, and c3 approach zero with A. In the limit as A
approaches zero the second of these equations becomes
(4) ^/»(0 + **"(0 + cy'(o = o.
453
454 SKEW CURVES [XI, §215
Hence the equation of the osculating plane is
(5) A(X-x) + B(Y-y-) + C(Z-z) = Q,
where A, B, and C satisfy the relations
(Adx + Bdy + C dz = 0 ,
Co) <
(Ad2x + Bd2y + Cd*z = 0.
The coefficients A, B, and C may be eliminated from (5) and (6),
and the equation of the osculating plane may be written in the form
X -x Y —y Z -z
dx dy dz
d?x d* d*z
= 0.
Among the planes which pass through the tangent, the osculating
plane is the one which the curve lies nearest near the point of tan-
gency. To show this, let us consider any other plane through the
tangent, and let F(f) be the function obtained by substituting
f(t + h), $(t -f h), \li(t + h) for X, F, Z, respectively, in the left-hand
side of the equation (5), which we shall now assume to be the equa
tion of the new tangent plane. Then we shall have
where 77 approaches zero with h. The distance from any second
point M' of F near M to this plane is therefore an infinitesimal of
the second order; and, since F(t) has the same sign for all sufficiently
small values of h, it is clear that the given curve lies wholly on one
side of the tangent plane considered, near the point of tangency.
These results do not hold for the osculating plane, however. For
that plane, Af" + B<f>" + Cif/" = 0 ; hence the expansions for the
coordinates of a point of F must be carried to terms of the third
order. Doing so, we find
h*
=
It follows that the distance from a point of T to the osculating
plane is an infinitesimal of the third order; and, since F(f) changes
sign with A, it is clear that a skew curve crosses its osculating plane
at their common point. These characteristics distinguish the oscu
lating plane sharply from the other tangent planes.
XI, §216] OSCULATING PLANE 455
216. Stationary osculating plane. The results just obtained are not
valid if the coefficients A, B, C of the osculating plane satisfy the
relation
(7) A dsx + Bd*y + Cd*z = 0.
If this relation is satisfied, the expansions for the coordinates must
be carried to terms of the fourth order, and we should obtain a
relation of the form
A d*x + B d*y + C d* z
The osculating plane is said to be stationary at any point of F for
which (7) is satisfied; if A d*x + Bd*y + Cd*z does not vanish
also, — and it does not in general, — F(t) changes sign with h and
the curve does not cross its osculating plane. Moreover the distance
from a point on the curve to the osculating plane at such a point is
an infinitesimal of the fourth order. On the other hand, if the
relation A d*x + Bd*y + Cd*z = 0 is satisfied at the same point,
the expansions would have to be carried to terms of the fifth order ;
and so on.
Eliminating A, B, and C between the equations (6) and (7), we
obtain the equation
dx dy dz
(8)
d*x d*y
dsz
= 0,
whose roots are the values of t which correspond to the points of F
where the osculating plane is stationary. There are then, usually,
on any skew curve, points of this kind.
This leads us to inquire whether there are curves all of whose
osculating planes are stationary. To be precise, let us try to find
all the possible sets of three functions x, y, z of a single variable t,
which, together with all their derivatives up to and including those
of the third order, are continuous, and which satisfy the equation
(8) for all values of t between two limits a and b (a < b).
Let us suppose first that at least one of the minors of A which
correspond to the elements of the third row, say dx d^y — dy d2x, does
not vanish in the interval (a, b). The two equations
( dz = Cl dx + C2 dy,
\'d*z = Cldix + C3<Py,
456 SKEW CURVES [XI, §216
which are equivalent to (6), determine C\ and C2 as continuous
functions of t in the interval (a, &). Since A = 0, these functions
also satisfy the relation
(10) d*z = C\dzx + Czd*y.
Differentiating each of the equations (9) and making use of (10),
we find
d€\ dx + dC2 dy = 0 , dC\ d'2x + dCz d*i/ = 0 ,
whence dC± = dC« = 0. It follows that each of the coefficients Cl
and C'2 is a constant ; hence a single integration of the first of
equations (9) gives
z — C\x + C2y + <73,
where Cs is another constant. This shows that the curve F is a
plane curve.
If the determinant dxd2y — dyd2x vanishes for some value c of the variable t
between a and b, the preceding proof fails, for the coefficients Ci and C% might
be infinite or indeterminate at such a point. Let us suppose for definiteness
that the preceding determinant vanishes for no other value of t in the interval
(a, 6), and that the analogous determinant dxd2z — dzd*x does not vanish for
t = c. The argument given above shows that all the points of the curve F which
correspond to values of t between a and c lie in a plane P, and that all the
points of F which correspond to values of t between c and 6 also lie in some
plane Q. But dxd^z — dzd*x does not vanish for t = c; hence a number h
can be found such that that minor does not vanish anywhere in the interval
(c — h, c + h). Hence all the points on T which correspond to values of t
between c — h and c + h must lie in some plane R. Since E must have an
infinite number of points in common with P and also with Q, it follows that
these three planes must coincide.
Similar reasoning shows that all the points of F lie in the same plane unless
all three of the determinants
dxd^y — dyd'2x, dxd2z — dzd?x, dyd^z — dzd^y
vanish at the same point in the interval (a, b). If these three determinants do
vanish simultaneously, it may happen that the curve F is composed of several
portions which lie in different planes, the points of junction being points at
which the osculating plane is indeterminate.*
If all three of the preceding determinants vanish identically in a certain
interval, the curve F is a straight line, or is composed of several portions of
straight lines. If dx/dt does not vanish in the interval (a, 6), for example, we
may write
d2zdx — dzd2x
whence
(dx)2
dy = Ci dx ,
(dx)*
dz — C2dx,
"This singular case seems to have been noticed first by Peano. It is evidently of
Interest only from a purely analytical standpoint.
XI, §217] OSCULATING PLANE 457
where C\ and C2 are constants. Finally, another integration gives
y = Cix+C"i, 2 = Caz+CJ,
which shows that T is a straight line.
217. Stationary tangents. The preceding paragraph suggests the study of
certain points on a skew curve which we had not previously defined, namely
the points at which we have
d2x = d*y = d*z
dx dy dz '
The tangent at such a point is said to be stationary. It is easy to show by the
formula for the distance between a point and a straight line that the distance
from a point of T to the tangent at a neighboring point, which is in general an
infinitesimal of the second order, is of the third order for a stationary tangent.
If the given curve T is a plane curve, the stationary tangents are the tangents at
the points of inflection. The preceding paragraph shows that the only curve
whose tangents are all stationary is the straight line.
At a point where the tangent is stationary, A = 0, and the equation of the
osculating plane becomes indeterminate. But in general this indetermination
can be removed. For, returning to the calculation at the beginning of § 215
and carrying the expansions of the coordinates of M ' to terms of the third order,
it is easy to show, by means of (11), that the equation of the plane through M'
and the tangent at M is of the form
X-x Y-y Z-z
f'(t) <t>'(t) y/(«)
= o,
where ti , e2 , c8 approach zero with h. Hence that plane approaches a perfectly
definite limiting position, and the equation of the osculating plane is given by
replacing the second of equations (6) by the equation
Bdsy
If the coordinates of the point M also satisfy the equation
d3x d3y dsz
dx dy dz
the second of the equations (6) should be replaced by the equation
Ad<ix = Bdiy + Cdvz = 0,
where q is the least integer for which this latter equation is distinct from the
equation A dx = Bdy + C dz = 0. The proof of this statement and the exami
nation of the behavior of the curve with respect to its osculating plane are left
to the reader.
Usually the preceding equation involving the third differentials is sufficient,
and the coefficients A, B, C do not satisfy the equation
Bd*y
In this case the curve crosses every tangent plane except the osculating plane.
458 SKEW CURVES [XI, §218
218. Special curves. Let us consider the skew curves T which satisfy a
relation of the form
(12) xdy-ydx =
where K is a given constant. From (12) we find immediately
* -ydsx + dxd*y - dyd*x = Kd3z.
Let us try to find the osculating plane of T which passes through a given point
(a, b, c) of space. The coordinates (x, ?/, z) of the point of tangency must satisfy
the equation
a — z b — y c — z
dx dy dz
= 0,
d2x d2y d*z
which, by means of (12) and (13), may be written in the form
(14) ay-bx + K(c-z) = Q.
Hence the possible points of tangency are the points of intersection of the
curve F with the plane (14), which passes through (a, 6, c).
Again, replacing dz, d?z and d3z by their values from (12) and (13), the equa
tion A = 0, which gives the points at which the osculating plane is stationary,
becomes
A = \ (dx d'2y - dy d2x)2 = 0 ;
hence we shall have at the same points
d'2x _ d*y _ 7/d2x - xd'2y _ d2z
dx dy ydx — xdy dz
which shows that the tangent is stationary at any point at which the osculating
plane is stationary.
It is easy to write down the equations of skew curves which satisfy (12) ; for
example, the curves
x = Atm, y = Bt», z = Ctm + n,
where A, B, C, m, and n are any constants, are of that kind. Of these
the simplest are the skew cubic x = t, y = i2, z — t3, and the skew quartic
x — t, y = t 3, z = t*. The circular helix
x = a cos t , y = a sin t , z = Kt
is another example of the same kind.
In order to find all the curves which satisfy (12), let us write that equation in
the form
d(xy — Kz) = 2ydx.
If we set
x =/(«), xy-Kz = 4>(t),
the preceding equation becomes
XI, §219] ENVELOPES OF SURFACES 459
Solving these three equations for z, y, and z, we find the general equations of T
in the form
(15) »=/(<), V = 1%L, Kz
'
where f(t) and <f>(t) are arbitrary functions of the parameter t. It is clear, how
ever, that one of these functions may be assigned at random without loss of
generality. In fact we may setf(t) = £, since this amounts to choosing/(£) as a
new parameter.
II. ENVELOPES OF SURFACES
Before taking up the study of the curvature of skew curves, we
shall discuss the theory of envelopes of surfaces.
219. One-parameter families. Let S be a surface of the family
(16) f(x,y,z,a)=0,
where a is the variable parameter. If there exists a surface E which
is tangent to each of the surfaces S along a curve C, the surface E
is called the envelope of the family (16), and the curve of tangency
C of the two surfaces S and E is called the characteristic curve.
In order to see whether an envelope exists it is evidently neces
sary to discover whether it is possible to find a curve C on each of
the surfaces S such that the locus of all these curves is tangent to
each surface S along the corresponding curve C. Let (x, y, z) be
the coordinates of a point M on a characteristic. If M is not a
singular point of S, the equation of the tangent plane to S at M is
df df df
dx <-Y -*) + ;£( r - y) + ± (*-«)- o.
i*1*^ ^y Cviv
As we pass from point to point of the surface E, x, y, z, and a are
evidently functions of the two independent variables which express
the position of the point upon E, and these functions satisfy the
equation (16). Hence their differentials satisfy the relation
(17) -/• dx + ~ du + -/- dz + J- da — 0.
OX cy cz va
Moreover the necessary and sufficient condition that the tangent
plane to E should coincide with the tangent plane to S is
dx dy dz
or, by (17),
W t? = 0.
460 SKEW CURVES [XI, §220
Conversely, it is easy to show, as we did for plane curves (§ 201),
that the equation R(x, y, z) = 0, found by eliminating the param
eter a between the two equations (16) and (18), represents one or
more analytically distinct surfaces, each of which is an envelope
of the surfaces S or else the locus of singular points of S, or a com
bination of the two. Finally, as in § 201, the characteristic curve
represented by the equations (16) and (18) for any given value of a
is the limiting position of the curve of intersection of S with a
neighboring surface of the same family.
220. Two-parameter families. Let S be any surface of the two-
parameter family
(19) f(x, y, z,a,V) =0,
where a and b are the variable parameters. There does not exist,
in general, any one surface which is tangent to each member of this
family all along a curve. Indeed, let b = <£(«) be any arbitrarily
assigned relation between a and b which reduces the family (19) to
a one-parameter family. Then the equation (19), the equation
b = <f>(a), and the equation
represent the envelope of this one-parameter family, or, for any
fixed value of a, they represent the characteristic on the correspond
ing surface S. This characteristic depends, in general, on <f>'(a),
and there are an infinite number of characteristics on each of the
surfaces S corresponding to various assignments of <£(a). There
fore the totality of all the characteristics, as a and b both vary arbi
trarily, does not, in general, form a surface. We shall now try to
discover whether there is a surface E which touches each of the
family (19) in one or more points, — not along a curve. If such a
surface exists, the coordinates (x, y, z) of the point of tangency of
any surface S with this envelope E are functions of the two variable
parameters a and b which satisfy the equation (19) ; hence their dif
ferentials dx, dy, dz with respect to the independent variables a
and b satisfy the relation
XI, §221] ENVELOPES OF SURFACES 461
Moreover, in order that the surface which is the locus of the point
of tangency (x, y, z) should be tangent to S, it is also necessary
that we should have
or, by (21),
Since a and b are independent variables, it follows that the equations
(22} — = 0 , -f- = 0
da cb
must be satisfied simultaneously by the coordinates (x, y, z~) of the
point of tangency. Hence we shall obtain the equation of the
envelope, if one exists, by eliminating a and b between the three
equations (19) and (22). The surface obtained will surely be tan
gent to S at (x, y, z) unless the equations
dx dy dz
are satisfied simultaneously by the values (x, y, z) which satisfy (19)
and (22) ; hence this surface is either the envelope or else the locus
of singular points of S.
We have seen that there are two kinds of envelopes, depending
on the number of parameters in the given family. For example,
the tangent planes to a sphere form a two-parameter family, and
each plane of the family touches the surface at only one point.
On the other hand, the tangent planes to a cone or to a cylinder
form a one-parameter family, and each member of the family is
tangent to the surface along the whole length of a generator.
221. Developable surfaces. The envelope of any one-parameter family
of planes is called a developable surface. Let
(23) z = ax + yf(«) + <K«)
be the equation of a variable plane P, where a is a parameter and
where /(a) and <£(«) are any two functions of a. Then the equa
tion (23) and the equation
(24) x + yf'(a) + *'(«) = 0
represent the envelope of the family, or, for a given value of a, they
represent the characteristic on the corresponding plane. But these
462 SKEW CURVES [XI, §221
two equations represent a straight line ; hence each characteristic
is a straight line G, and the developable surface is a ruled surface.
We proceed to show that all the straight lines G are tangent to the
same skew curve. In order to do so let us differentiate (24) again
with regard to a. The equation obtained
(25) y/»(a) + <»» = 0
determines a particular point M on G. We proceed to show that G
is tangent at M to the skew curve F which M describes as a varies.
The equations of F are precisely (23), (24), (25), from which, if we
desired, we might find x, y, and z as functions of the variable
parameter a. Differentiating the first two of these and using the
third of them, we find the relations
(26) dz = a dx + /(a) d>j , dx + /'(«) dy = 0 ,
which show that the tangent to F is parallel to G. But these two
straight lines also have a common point ; hence they coincide.
The osculating plane to the curve F is the plane P itself. To
prove this it is only necessary to show that the first and second
differentials of x, y, and z with respect to a satisfy the relations
dz = a dx
The first of these is the first of equations (26), which is known to
hold. Differentiating it again with respect to a, we find
d*z = ad*x + f(a}d'iy + [dx + f(a)dy']da,
which, by the second of equations (26), reduces to the second of the
equations to be proved.
It follows that any developable surface may be defined as the locus
of the tangents to a certain skew curve T. In exceptional cases the
curve F may reduce to a point at a finite or at an infinite distance ;
then the surface is either a cone or a cylinder. This will happen
whenever /"(a) = 0.
Conversely, the locus of the tangents to any skew curve F is a
developable surface. For, let
be the equations of any skew curve F. The osculating planes
y) + C(Z - z) = 0
XI, §221] ENVELOPES OF SURFACES 463
form a one-parameter family, whose envelope is given by the pre
ceding equation and the equation
dA(X - x) + dB(Y - y) + dC(Z - 2) = 0 .
For any fixed value of t the same equations represent the charac
teristic in the corresponding osculating plane. We shall show that
this characteristic is precisely the tangent at the corresponding
point of F. It will be sufficient to establish the equations
A dx + Bdy + C dz = 0, dA dx + dB dy -f dC dz = 0 .
The first of these is the first of (6), while the second is easily
obtained by differentiating the first and then making use of the
second of (6). It follows that the characteristic is parallel to
the tangent, and it is evident that each of them passes through
the point (x, y, z) ; hence they coincide.
This method of forming the developable gives a clear idea of
the appearance of the surface. Let AB be an arc of a skew curve.
At each point M of AB draw the tangent, and consider only that
half of the tangent which extends in a certain direction, — from A
toward B, for example. These half rays form one nappe Si of the
developable, bounded on three sides by the arc AB and the tan
gents A and B and extending to infinity. The other ends of the tan
gents form another nappe S2 similar to Si and joined to Si along the
arc AB. To an observer placed above them these two nappes appear
to cover each other partially. It is evident that any plane not tan
gent to F through any point 0 of AB cuts the two nappes Si and S2
of the developable in two branches of a curve which has a cusp at O.
The skew curve F is often called the edge of regression of the
developable surface.*
It is easy to verify directly the statement just made. Let us
take O as origin, the secant plane as the xy plane, the tangent to F as
the axis of 2, and the osculating plane as the xz plane. Assuming
that the coordinates x and y of a point of F can be expanded in powers
of the independent variable 2, the equations of F are of the form
x = azz2 + a3z* -\ , y = b3z* + •••,
for the equations
dx _ dy _ dzy _
dz dz dz*
* The English term " edge of regression " does not suggest that the curve is a locus
of cusps. The French terms "arete de rebroussement " and "point de rebroussement "
are more suggestive. — TRANS.
464 SKEW CURVES [XI §222
must be satisfied at the origin. Hence the equations of a tangent
at a point near the origin are
Setting Z = 0, the coordinates .Y and F of the point where the tan
gent meets the secant plane are found to have developments which
begin with terms in z2 and in «8, respectively ; hence there is surely
a cusp at the origin.
Example. Let us select as the edge of regression the skew cubic x — t, y = t2,
z = t8. The equation of the osculating plane to the curve is
(27) *»-3«aJr + 3tr-Z = 0;
hence we shall obtain the equation of the corresponding developable by writing
down the condition that (27) should have a double root in t, which amounts to
eliminating t between the equations
*- Z = 0.
The result of this elimination is the equation
(XY - Z)2 - 4(X2 - F)(F2 - JTZ) = 0,
which shows that the developable is of the fourth order.
It should be noticed that the equations (28) represent the tangent to the given
cubic.
222. Differential equation of developable surfaces. If z = F(x, y) be
the equation of a developable surface, the function F(x, y) satisfies
the equation s2 — rt = 0, where r, s, and t represent, as usual, the
three second partial derivatives of the function F(x, y).
For the tangent planes to the given surface,
Z =pX + qY + z — px — qy,
must form a one-parameter family ; hence only one of the three
coefficients p, q, and z — px — qy can vary arbitrarily. In particular
there must be a relation between p and q of the form f(p, q) = 0.
It follows that the Jacobian D( p, q)/D(x, y) = rt — s2 must vanish
identically.
Conversely, if F(x, y) satisfies the equation rt — s2 = 0, p and q
are connected by at least one relation. If there were two distinct
relations, p and q would be constants, F(x, y) would be of the form
ax -f by + c, and the surface z = F(x, y) would be a plane. If there
XI, §223] ENVELOPES OF SURFACES 465
is a single relation between p and q, it may be written in the form
a = f(p\ where p does not reduce to a constant. But we also have
v (rt _ n - D(*-px-qy,p\
y(rt
hence z—px — qy is also a function of p, say *l/(p), whenever
rt — s2 = 0. Then the unknown function F(x, y) and its partial
derivatives p and q satisfy the two equations
Differentiating the second of these equations with respect to x and
with respect to ?/, we find
ff = °» L* + ^W + f
Since p does not reduce to a constant, we must have
hence the equation of the surface is to be found by eliminating p
between this equation and the equation
which is exactly the process for finding the envelope of the family
of planes represented by the latter equation, p being thought of as
the variable parameter.
223. Envelope of a family of skew curves. A one-parameter family
of skew curves has, in general, no envelope. Let us consider first
a family of straight lines
(29) x = az+p, y = bz + q,
where a, b, p, and q are given functions of a variable parameter a.
We shall proceed to find the conditions under which every member
of this family is tangent to the same skew curve T. Let z = <£(a)
be the z coordinate of the point M at which the variable straight
line D touches its envelope T. Then the required curve T will be
represented by the equations (29) together with the equation
z — <£(<*), an(l the direction cosines of the tangent to T will be pro
portional to dx/da, dy/da, dz/da, i.e. to the three quantities
a <£'(«) + «'*(«) +JP', b<j>'(a)+b'<l>(a) + q', ^'(a),
466 SKEW CURVES [XI, §223
where a', b', p', and q' are the derivatives of a, b, p, and q, respec
tively. The necessary and sufficient condition that this tangent be
the straight line D itself is that we should have
dx _ dz_ d_y _ dz
da da da da
that is,
The unknown function <£(a) must satisfy these two equations;
hence the family of straight lines has no envelope unless the two
are compatible, that is, unless
a'q'-b'p' = 0.
If this condition is satisfied, we shall obtain the envelope by setting
<t>(a}=-p'/a<=-q</b>.
It is easy to generalize the preceding argument. Let us consider a
one-parameter family of skew curves (C) represented by the equations
(30) F(x, y, a, a) = 0 , *(x, y,z,a) = Q,
where a is the variable parameter. If each of these curves C is
tangent to the same curve T, the coordinates (x, y, z) of the point
M at which the envelope touches the curve C which corresponds to
the parameter value a are functions of a which satisfy (30) and
which also satisfy another relation distinct from those two. Let
dx, dy, dz be the differentials with respect to a displacement of M
along C ; since a is constant along C, these differentials must satisfy
the two equations
dF dF dF J
-j- dx + -7— dy + -^- dz = 0,
dx *y **
0* , ^ a* A
-dx + -^-dy + —dz = Q.
dy cz
On the other hand, let &r, 8y, &s, 8a be the differentials of x, y, z,
and a with respect to a displacement of M along T. These differen
tials satisfy the equations
(32)
cy
XI, §223] ENVELOPES OF SURFACES 467
The necessary and sufficient conditions that the curves C and T
be tangent are
dx _ dy _ dz
~Sx~ 8~y ~~ ~Sz'
or, making use of (31) and (32),
It follows that the coordinates (x, y, z) of the point of tangency must
satisfy the equations
dF d&
(33) F-0, * = 0, ^ = 0, ^ = 0.
Hence, if the family (30) is to have an envelope, the four equations
(33) must be compatible for all values of a. Conversely, if these
four equations have a common solution in x, y, and z for all values
of a, the argument shows that the curve T described by the point
(x, y, z) is tangent at each point (x, y, z) upon it to the correspond
ing curve C. This is all under the supposition that the ratios between
dx, dy, and dz are determined by the equations (31), that is, that the
point (x, y, 2) is not a singular point of the curve C.
Note. If the curves C are the characteristics of a one-parameter
family of surfaces F(x, y, z, a) = 0, the equations (33) reduce to
the three distinct equations
(34) F.p, |f = 0,
hence the curve represented by these equations is the envelope
of the characteristics. This is the generalization of the theorem
proved above for the generators of a developable surface.
The equations of a one-parameter family of straight lines are often written
in the form
(35) x -x0 _y -yo_z - z0 ^
a b c
where XQ , yo , Zo , o, &, c are functions of a variable parameter a. It is easy to
find directly the condition that this family should have an envelope. Let I
denote the common value of each of the preceding ratios ; then the coordinates
of any point of the straight line are given by the equations
x = x0 + la , y = y0 + lb , z = ZQ + Ic ,
and the question is to determine whether it is possible to substitute for I such a
function of a that the variable straight line should always remain tangent to
468
SKEW CURVES
[XI, § 224
the curve described by the point (z, y, z). The necessary condition for this is
that we should have
,q<n Xp + g'Z _ 2/6 + b'l 26 + c'l
^oo; — _ .
a b c
Denoting by m the common value of these ratios and eliminating I and m from
the three linear equations obtained, we find the equation of condition
(37)
a' b' c'
= 0.
If this condition is satisfied, the equations (36) determine I, and hence also the
equation of the envelope.
III. CURVATURE AND TORSION OF SKEW CURVES
224. Spherical indicatrix. Let us adopt upon a given skew curve F
a definite sense of motion, and let s be the length of the arc AM
measured from some fixed point A as origin to any point M, affixing
the sign -f or the sign — according as the direction from A toward
M is the direction adopted or the opposite direction. Let MT be
the positive direction of the tangent at M, that is, that which cor
responds to increasing values of the arc. If through any point O in
space lines be drawn parallel to these half rays, a cone S is formed
which is called the directing cone of the developable surface formed
by the tangents to F. Let us draw a sphere of unit radius about O
as center, and let 2 be the line of intersection of this sphere with
the directing cone. The curve 2 is called the spherical indicatrix
FIG. 48
of the curve F. The correspondence between the points of these two
curves is one-to-one : to a point M of F corresponds the point m where
the parallel to MT pierces the sphere. As the point M describes the
CURVATURE TORSION 469
curve F in the positive sense, the point m describes the curve 2 in
a certain sense, which we shall adopt as positive. Then the corre
sponding arcs s and o- increase simultaneously (Fig. 48).
It is evident that if the point O be displaced, the whole curve 2
undergoes the same translation ; hence we may suppose that 0 lies
at the origin of coordinates. Likewise, if the positive sense on the
curve F be reversed, the curve 2 is replaced by a curve symmetrical
to it with respect to the point 0 ; but it should be noticed that the
positive sense of the tangent mt to 2 is independent of the sense of
motion on T.
The tangent plane to the directing cone along the generator Om is
parallel to the osculating plane at M. For, let AX + BY + CZ = 0
be the equation of the plane Omm', the center 0 of the sphere being
at the origin. This plane is parallel to the two tangents at M and
at M ' ; hence, if t and t + h are the parameter values which corre
spond to M and M'} respectively, we must have
(38) Af'(t) + BV(t) + Cf (0 = 0.
(39) Af(t + A) + B#(t + k) + Cf (* + A) = 0.
The second of these equations may be replaced by the equation
*) -
B + c = 0
h h h
which becomes, in the limit as h approaches zero,
(40) Af"(£) + B<j>"(£) + Cf'09 = 0.
The equations (38) and (40), which determine A, B, and C for the
tangent plane at m, are exactly the same as the equations (6) which
determine A, B, and C for the osculating plane.
225. Radius of curvature. Let o> be the angle between the positive
directions of the tangents MT and M'T' at two neighboring points
M and M' of F. Then the limit of the ratio w/arc MM', as M
approaches M', is called the curvature of F at the point M, just as
for a plane curve. The reciprocal of the curvature is called the
radius of curvature ; it is the limit of arc MM' /to.
Again, the radius of curvature R may be defined to be the limit
of the ratio of the two infinitesimal arcs MM' and mm', for we have
arc MM' arc MM' arc mm' chord mm'
VX
arc mm' chord mm'
470 SKEW CURVES [XI, §225
and each of the fractions (arc raw')/(chord) mm' and (chord mm')/<a
approaches the limit unity as m approaches m'. The arcss( = 3/Af')
and o-(=mm') increase or decrease simultaneously; hence
(«> *=£•
Let the equations of T be given in the form
(42) x=f(t), y = *(0, * = lKO»
where 0 is the origin of coordinates. Then the coordinates of the
point m are nothing else than the direction cosines of MT, namely
dx dy dz
a — — j a — — > y = 3 —
efo ete ds
Differentiating these equations, we find
- dx d2s ds d*y - dy d*s
ri - ' dB= - ,2 - '
s2 ds2
where O indicates as usual the sum of the three similar terms
obtained by replacing x by x, y, z successively. Finally, expanding
and making use of the expressions for ds2 and ds d2s, we find
. Sdx*
ds4
By Lagrange's identity (§ 131) this equation may be written in
the form
, 2 A* + B* + C*
*° - -*T
where
!A = dyd?z — dzd^y, B = dzdzx — dxd*z,
C =
a notation which we shall use consistently in what follows. Then
the formula (41) for the radius of curvature becomes
and it is evident that Rz is a rational function of x, y, z, x', y', z',
x", y", z". The expression for the radius of curvature itself is
irrational, but it is essentially a positive quantity.
XI, §226] CURVATURE TORSION 471
Note. If the independent variable selected is the arc s of the
curve r, the functions /(*), <£(«), and ^(s) satisfy the equation
/"(*) + *"(«) + *"(«)=!•
Then we shall have
(45)
«=/'(*)» ft = *'(•), y =
« =/"(*) «fe, dp = 4>"(s)ds, dy =
and the expression for the radius of curvature assumes the partic
ularly elegant form
(44/) ^ = [/"(«)]' + [*"(*)? + Cf («)]*•
226. Principal normal. Center of curvature. Let us draw a line
through M (on T) parallel to w£, the tangent to 2 at m. Let MN
be the direction on this line which corresponds to the positive direc
tion mt. The new line MN is called the principal normal to T at M :
it is that normal which lies in the osculating plane, since mt is
perpendicular to Om and Omt is parallel to the osculating plane
(§ 224). The direction MN is called the positive direction of the
principal normal. This direction is uniquely defined, since the posi
tive direction of mt does not depend upon the choice of the positive
direction upon T. We shall see in a moment how the direction in
question might be defined without using the indicatrix.
If a length MC equal to the radius of curvature at M be laid off
on MN from the point A/, the extremity C is called the center of
curvature of T at M, and the circle drawn around C in the osculat
ing plane with a radius MC is called the circle of curvature. Let
a'> ft'> y' be the direction cosines of the principal normal. Then the
coordinates (aju ylt z-^ of the center of curvature are
But we also have
a' = ^_^^i_ p^.— » dsdzx — dxdzs
da- ds da- ds dsa
and similar formulae for /?' and y'. Replacing a' by its value in
the expression for x, we find
^da&x
x^x + R2- -7^
ds3
472 SKEW CURVES
But the coefficient of R2 may be written in the form
d*x Sdx*-dx
[XI, §226
ds* ds*
or, in terms of the quantities A, B, and C,
Bdz — Cdy
ds*
The values of yl and zl may be written down by cyclic permutation
from this value of xl , and the coordinates of the center of curvature
may be written in the form
(46)
B dz —
C dx — A dz
ds*
A dii — B dx
These expressions for xlt y:, and zv are rational in x, y, z, x', y', z',
x" y" z".
A plane Q through M perpendicular to MN passes through the
tangent MT and does not cross the curve T at M. We shall proceed
to show that the center of curvature and the points of T near M lie
on the same side of Q. To show this, let us take as the independent
variable the arc s of the curve T counted from M as origin. Then
the. coordinates X, Y, Z of a, point M' of T near M are of the form
s dx s2
the expansions for Y and Z being similar to the expansion for X.
But since s is the independent variable, we shall have
dx
ds
dzx
ds2
da da d<r _ 1 ,
ds da- ds R
and the formula for A' becomes
If in the equation of the plane Q,
a'(X - *) + J8'(F - y)
1.2
XI, §227] CURVATURE TORSION 473
A", F, and Z be replaced by these expansions in the left-hand member,
the value of that member is found to be
I <«• + W + rf) + 0(5 + ') = I' (! +
where t] approaches zero with s. This quantity is positive for all
values of s near zero. Likewise, replacing (X, Y, Z} by the coordi
nates (x + Ra', y + Rfi', z + .Ry') of the center of curvature, the
result of the substitution is R, which is essentially positive. Hence
the theorem is proved.
227. Polar line. Polar surface. The perpendicular A to the oscu
lating plane at the center of curvature is called the polar line. This
straight line is the characteristic of the normal plane to T. For, in
the first place, it is evident that the line of intersection D of the
normal planes at two neighboring points M and M ' is perpendicular
to each of the lines MT and M'T' ; hence it is also perpendicular to
the plane mOm'. As M' approaches M, the plane mOm' approaches
parallelism to the osculating plane ; hence the line D approaches a
line perpendicular to the osculating plane. On the other hand, to
show that it passes through the center of curvature, let s be the
independent variable ; then the equation of the normal plane is
(47) a(X -x) + fi(Y-y) + y(Z -*)«*«,
and the characteristic is denned by (47) together with the equation
(48) | (X - x) + | ( Y - y} + £ (Z - ») - 1 = 0 .
This new equation represents a plane perpendicular to the principal
normal through the center of curvature ; hence the intersection of
the two planes is the polar line.
The polar lines form a ruled surface, which is called the polar
surface. It is evident that this surface is a developable, since we
have just seen that it is the envelope of the normal plane to F.
If F is a plane curve, the polar surface is a cylinder whose right
section is the evolute of F ; in this special case the preceding state
ments are self-evident.
228. Torsion. If the words "tangent line" in the definition of
curvature (§ 225) be replaced by the words " osculating plane," a
new geometrical concept is introduced which measures, in a manner,
the rate at which the osculating plane turns. Let a>' be the angle
between the osculating planes at two neighboring points M and M ';
474 SKEW CURVES [XI, §228
then the limit of the ratio o>'/arc MM', as M approaches M', is called
the torsion of the curve F at the point M. The reciprocal of the
torsion is called the radius of torsion.
The perpendicular to the osculating plane at M is called the
binormal. Let us choose a certain direction on it as positive, — we
shall determine later which we shall take, — and let a", ft", y" be
the corresponding direction cosines. The parallel line through the
origin pierces the unit sphere at a point n, which we shall now put
into correspondence with the point M of T. The locus of n is a
spherical curve ®, and it is easy to show, as above, that the radius
of torsion T may be defined as the limit of the ratio of the two corre
sponding arcs MM' and nn' of the two curves T and ®. Hence we
shall have
1--^,
dr2
where T denotes the arc of the curve ®.
The coordinates of n are a", ft", y", which are given by the formulae
(§ 215)
q"= . A => ft" = —=£=, y"=
where the radical is to be taken with the same sign in all three
formulae. From these formulae it is easy to deduce the values of
da", dp", dy"; for example,
da" = ± (X2 + -B2 + C*)dA—A(AdA + BdB + CdC)
(A2 + B2 + C2)f
whence, since dr2 = da"2 + dft"2 + dy"2,
-m SA* S**-IS(***)7
(A2 + B2+ C2)2
or, by Lagrange's identity,
S^dC-CdB}*
(A2 + B2 + C2)2 '
where & denotes the sum of the three terms obtained by cyclic per
mutation of the three letters A, B, C. The numerator of this expres
sion may be simplified by means of the relations
Adx+ Bdy + C dz = 0,
dA dx -f- dB dy + dC dz = 0,
whence
dx d d* *
B dC - C dB CdA - AdC A dB - B dA K
XI, §228] CURVATURE TORSION 475
where K is a quantity defined by the equation (49) itself. This gives
2 _ A'W
~ (A2 + B* + C2)2'
where K is defined by (49) ; or, expanding,
dz dx
dx dy
dsx day
d*y),
dx dy
dz dx
d*z dax
where o denotes the sum of the three terms obtained by cyclic per
mutation of the three letters x, y, z. But this value of K is exactly
the development of the determinant A [(8), § 216]; hence
- A* + B* + C2
and therefore the radius of torsion is given by the formula
^2_|_^2_j_ ^2
(50)
T=±
If we agree to consider T essentially positive, as we did the radius
of curvature, its value will be the absolute value of the second mem
ber. But it should be noticed that the expression for T is rational
in x, y, z, x', y', z', x", y", z" ; hence it is natural to represent the
radius of torsion by a length affected by a sign. The two signs
which T may have correspond to entirely different aspects of the
curve F at the point M.
Since the sign of T depends only on that of A, we shall investigate
the difference in the appearance of F near M when A has different
signs. Let us suppose that the trihedron Oxyz is placed so that an
observer standing on the xy plane with his feet at 0 and his head in
the positive z axis would see the x axis turn through 90° to his left
if the x axis turned round into the y axis (see footnote, p. 477).
Suppose that the positive direction of the binormal MNb has been so
chosen that the trihedron formed from the lines MT, MN, MNb has
the same aspect as the trihedron formed from the lines Ox, Oy, Oz ;
that is, if the curve F be moved into such a position that M coincides
with O, MT with Ox, and MN with Oy, the direction MNb will coin
cide with the positive z axis. During this motion the absolute value
of T remains unchanged ; hence A cannot vanish, and hence it cannot
176 SKEW CURVES [XI, § 228
even change sign.* In this position of the curve T with respect to
the axes now in the figure the coordinates of a point near the origin
will be given by the formulae
f* =
(51) \y =
where e, e', e" approach zero with t, provided that the parameter t is
so chosen that t = 0 at the origin. For with the system of axes
employed we must have dy = dz = d2z = 0 when t = 0. Moreover
we may suppose that at > 0, for a change in the parameter from t to
— t will change at to — at . The coefficient &2 is positive since y must
be positive near the origin, but cz may be either positive or negative.
On the other hand, f or t = 0, A = 12a1J2c3 dt*. Hence the sign of A
is the sign of e3. There are then two cases to be distinguished. If
c3 > 0, x and z are both negative for — h < t < 0, and both positive
f or 0 < t < h, where h is a sufficiently small positive number ; i.e.
an observer standing on the xy plane with his feet at a point P on
y
M'
,-'' M M "X
/ \
I'M" ^v'
FIG. 49, a FIG. 49, 6
the positive half of the principal normal would see the arc MM' at
his left and above the osculating plane, and the arc MM" at his right
below that plane (Fig. 49, a). In this case the curve is said to be
sinistrorsal. On the other hand, if c3 < 0, the aspect of the curve
would be exactly reversed (Fig. 49, b), and the curve would be said
to be dextrorsal. These two aspects are essentially distinct. For
example, if two spirals (helices) of the same pitch be drawn on the
same right circular cylinder, or on two congruent cylinders, they
will be superposable if they are both sinistrorsal or both dextrorsal ;
but if one of them is sinistrorsal and the other dextrorsal, one of
them will be superposable upon the helix symmetrical to the other
one with respect to a plane of symmetry.
* It would be easy to show directly that A does not change sign when we pass from
one set of rectangular axes to another set which have the same aspect.
XI, §229] CURVATURE TORSION 477
In consequence of these results we shall write
(52) r__4i±*±<?;
i.e. at a point where the curve is dextrorsal T shall be positive, while
T shall be negative at a point where the curve is sinistrorsal. A dif
ferent arrangement of the original coordinate trihedron Oxyz would
lead to exactly opposite results.*
229. Frenet's formulae. Each point M of T is the vertex of a tri-
rectangular trihedron whose aspect is the same as that of the trihe
dron Oxyz, and whose edges are the tangent, the principal normal,
and the binomial. The positive direction of the principal normal is
already fixed. That of the tangent may be chosen at pleasure, but
this choice then fixes the positive direction on the binormal. The dif
ferentials of the nine direction cosines (a, ft, y), (a1, ft', •/). (a", ft", y")
of these edges may be expressed very simply in terms of R, T, and
the direction cosines themselves, by means of certain formulae due
to Frenet.f We have already found the formulae for da, dft, and dy :
/KQN da a' dft ft' dy y'
( oo ) — = — > — *- = — > — '- = i— •
ds R ds R ds R
The direction cosines of the positive binormal (§ 228) are
~"-- A B " = C
where e = ± 1. Since the trihedron (MT, MN, MNb) has the same
aspect as the trihedron Oxyz, we must have
or „• =
l2 + B2 + C2
On the other hand, the formula for da" may be written
d „ _ B(B dA -AdB} + C(C dA - A dC)
(A2 + B2 + C2)*
or, by (49) and the relation A' = A,
da" = ^ Cft-By q'A
ds (A2 + B2 + C'2)3 ~ ^ + &+ C12'
* It is usual in America to adopt an arrangement of axes precisely opposite to that
described above. Hence we should write T = + (A* + B^ + C'2)/A, etc. See also
the footnote to formula (54), §229. — TRANS.
t Nouvelles Annales de Mathematiques, 1864, p. 281.
478 SKEW CURVES [XI, §229
The coefficient of a' is precisely 1/T, by (52). The formulae for
dp" and dy" may be calculated in like manner, and we should find
'
ds T ds T ds T
which are exactly analogous to (53).*
In order to find da', d(3', dy', let us differentiate the well-known
formulae
«'2+ (3'2+ y'a = l,
cm' + /3/3' + yy' = 0,
replacing da, d(3, dy, da", d/3", dy" by their values from (53) and
(54). This gives
a' da' + ft' dp + y' dy = 0,
ds
a da' + J3 d/3' + y dy' + — = 0,
whence, solving for da', dft', dy',
ds
-
ds R T ds ' R T ds R T
The formulae (53), (54), and (55) constitute Frenet's formulae.
Note. The formulae (54) show that the tangent to the spherical
curve ® described by the point n whose coordinates are a", /?", y" is
parallel to the principal normal. This can be verified geometrically.
Let S' be the cone whose vertex is at 0 and whose directrix is the
curve ®. The generator On is perpendicular to the plane which is
tangent to the cone S along Om (§ 228). Hence S' is the polar cone
to S. But this property is a reciprocal one, i.e. the generator Om
of 5 is surely perpendicular to the plane which is tangent to .S"
along On. Hence the tangent mt to the curve 2, since it is perpen
dicular to each of the lines On and Om, is perpendicular to the
plane mOn. For the same reason the tangent nt' to the curve ® is
perpendicular to the plane mOn. It follows that mt and nt' are
parallel.
* If we had written the formula for the torsion in the form l/T= A/ (A* + B* + C'2),
Frenet's formulae would have to be written in the form da" /ds = — a'/T, etc.
[Hence this would be the form if the axes are taken as usual in America. — TRANS.]
XI, § 230]
CURVATURE TORSION
479
230. Expansion of x, y, and z in powers of s. Given two functions
R = <£(s), T = \l/(s) of an independent variable s, the first of which
is positive, there exists a skew curve T which is completely defined
except for its position in space, and whose radius of curvature and
radius of torsion are expressed by the given equations in terms of
the arc s of the curve counted from some fixed point upon it. A rig
orous proof of this theorem cannot be given until we have discussed
the theory of differential equations. Just now we shall merely show
how to find the expansions for the coordinates of a point on the
required curve in powers of s, assuming that such expansions exist.
Let us take as axes the tangent, the principal normal, and the
binormal at O, the origin of arcs on T. Then we shall have
(56)
s
*=I
dx
+
= i (*y\
1 \C&/I
+
o^z\
+
/dsx\
-ri) +
£') +
1.2 VdsVo
s2 /<?_»•
172 Vo^
+
123 \ds3/ o
1.2.3 W/o+
g3 /rf8 ^\
1.2.3\Ai/t
a'
ll
where #, y, and « are the coordinates of a point on F. But
dx dzx da
— = a, -r-T = -7-
as as as
whence, differentiating,
d*x _ _ a' dR 1 la
~d&~ ~ R2 ds R \R "' T,
In general, the repeated application of Frenet's formulae gives
where Ln, Mn, Pn are known functions of R, T, and their successive
derivatives with respect to s. In a similar manner the successive
derivatives of y and z are to be found by replacing (a, a', a") by
(/3, /?', /8") and (y, y', y"), respectively. But we have, at the origin,
a0 = 1, ft = 0, y0 - 0, a'n = 0, $ = 1,^ = 0, a'0' = 0, $' = 0, ^ = 1 ;
hence the formulas (56) become
(56')
.s3 dR
2R BR* ds
QRT
480 SKEW CURVES [XI, §231
where the terms not written down are of degree higher than three.
It is understood, of course, that R, T, dR/ds, • • • are to be replaced
by their values for 5 = 0.
These formulae enable us to calculate the principal parts of cer
tain infinitesimals. For instance, the distance from a point of the
curve to the osculating plane is an infinitesimal of the third order,
and its principal part is — s3/6RT. The distance from a point on
the curve to the x axis, i.e. to the tangent, is of the second order,
and its principal part is s*/2R (compare § 214). Again, let us cal
culate the length of an infinitesimal chord c. We find
,
where the terms not written down are of degree higher than four.
This equation may be written in the form
which shows that the difference s — c is an infinitesimal of the
third order and that its principal part is s8/24#2.
In an exactly similar manner it may be shown that the shortest
distance between the tangent at the origin and the tangent at a
neighboring point is an infinitesimal of the third order whose prin
cipal part is ss/12RT. This theorem is due to Bouquet.
231. Involutes and evolutes. A curve I\ is called an involute of a
second curve F if all the tangents to F are among the normals to I\,
and conversely, the curve F is called an evolute of F^ It is evident
that all the involutes of a given curve F lie on the developable sur
face of which F is the edge of regression, and cut the generators of
the developable orthogonally.
Let (x, y, z) be the coordinates of a point M of F, (a, ft, y) the
direction cosines of the tangent MT, and I the segment MMi between
M and the point Ma where a certain involute cuts MT. Then the
coordinates of Ml are
Xl=X + la, 7/1=:7/ + /yS, Kl = z + ly,
whence
dxl = dx + Ida + a dl,
dyi = di/ + l</(3 + (3 dl,
dzi = dz + I dy + y dl.
XI, § 231]
CURVATURE TORSION
481
In order that the curve described by Afx should be normal to
it is necessary and sufficient that a dx^ + (3 dyl + y dz± should vanish,
i.e. that we should have
a
dx
(ldy + ydz + dl + l(ada + {3dp + y dy) = 0,
which reduces to ds -\- dl = 0. It follows that the involutes to a
given skew curve F may be drawn by the same construction which
was used for plane curves (§ 206).
Let us try to find all the evolutes of a
given curve F, that is, let us try to pick
out a one-parameter family of normals to
the given curve according to some contin
uous law which will group these normals
into a developable surface (Fig. 50). Let
D be an evolute, <£ the angle between the
normal MM^ and the principal normal MN,
and I the segment MP between M and the
projection P of the point Ml on the principal normal. Then the
coordinates (xi} yi, z^) of Ml are
FIG. 50
(57)
£«" tan <£,
/£" tan <£,
ly" tan <£,
as we see by projecting the broken line MPM^ upon the three axes
successively. The tangent to the curve described by the point M^
must be the line MMt itself, that is, we must have
dxl
dzl
Let k denote the common value of these ratios ; then the condition
dxl = k(xl — x) may be transformed, by inserting the values of x^
and dxl and applying Frenet's formulae, into the form
a ds 1
\
- 4) + a' (
K/ \
+ a" d(l
— - kl)
T I
tan
- — ^ - kl tan
> =0.
The conditions dy^ = ]c(y^ — y) and dzl = k(zl — z) lead to exactly
similar forms, which may be deduced from the preceding by repla
cing (a, a', a") by ((3, /3', £") and (y, y', y"), respectively. Since the
482 SKEW CURVES [XI, §231
determinant of the nine direction cosines is equal to unity, these
three equations are equivalent to the set
(58)
ds
dl + I tan <f> — = kl,
Ids
d(l tan <£) — = kl tan <f> .
From the first of these I = R, which shows that the point P is the
center of curvature and that the line PM is the polar line. It fol
lows that all the evolutes of a given skew curve T lie on the polar sur
face. In order to determine these evolutes completely it only remains
to eliminate k between the last two of equations (58). Doing so
and replacing I by R throughout, we find ds = T d<j>. Hence <£ may
be found by a single quadrature :
(59) + -*
If we consider two different determinations of the angle <£ which
correspond to two different values of the constant <£0, the difference
between these two determinations of <£ remains constant all along T.
It follows that two normals to the curve T which are tangent to two
different evolutes intersect at a constant angle. Hence, if we know
a single family of normals to T which form a developable surface,
all other families of normals which form developable surfaces may
be found by turning each member of the given family of normals
through the same angle, which is otherwise arbitrary, around its
point of intersection with T.
Note L If T is a plane curve, T is infinite, and the preceding
formula gives <f> = <£„. The evolute which corresponds to <£0 = 0 is
the plane evolute studied in § 206, which is the locus of the centers
of curvature of F. There are an infinite number of other evolutes,
which lie on the cylinder whose right section is the ordinary evo
lute. We shall study these curves, which are called helices, in the
next section. This is the only case in which the locus of the cen
ters of curvature is an evolute. In order that (59) should be satis
fied by taking <£ = 0, it is necessary that T should be infinite or
that A should vanish identically ; hence the curve is in any case a
plane curve (§ 216).
XI, §232] CURVATURE TORSION 483
Note II. If the curve D is an evolute of T, it follows that T is an
involute of D. Hence
dst = d(MMl) ,
where sx denotes the length of the arc of the evolute counted from
some fixed point. This shows that all the evolutes of any given
curve are rectifiable.
232. Helices. Let C be any plane curve and let us lay off on the perpendic
ular to the plane of C erected at any point m on C a length mM proportional to
the length of the arc a- of C counted from some fixed point A. Then the skew
curve F described by the point M is called a helix. Let us take the plane of C
as the xy plane and let
z=/(o-), y = <t>(v)
be the coordinates of a point TO of C in terms of the arc o". Then the coordi
nates of the corresponding point M of the curve F will be
(60) x=/(<r), y = *(cr), z = K<r,
where K is the given factor of proportionality. The functions / and <f> satisfy
the relation /'2 + <£'2 = 1 ; hence, from (60),
where s denotes the length of the arc of T. It follows that s = cr Vl + K2 + J7,
or, if s and a- be counted from the same point A on C, s = <r Vl + -K2, since H = 0.
The direction cosines of the tangent to F are
(61) a
Since y is independent of (r, it is evident that the tangent to F makes a constant
angle with the z axis ; this property is characteristic : Any curve whose tangent
makes a constant angle with a fixed straight line is a helix. In order to prove
this, let us take the z axis parallel to the given straight line, and let C be the
projection of the given curve F on the xy plane. The equations of F may always
be written in the form
(62) x =/(<r) , y = <t>(v), z = ^(<r) ,
where the functions / and </> satisfy the relation /'2 4- <£'2 = 1, for this merely
amounts to taking the arc o- of C as the independent variable. It follows that
dS •v/V/2 4- rf/2 _1_ 1/~1
hence the necessary and sufficient condition that y be constant is that f should
be constant, that is, that \f/(a-) should be of the form Kcr + z0. It follows that
the equations of the curve F will be of the form (60) if the origin be moved to
the point x = 0, y = 0, z = z0.
Since y is constant, the formula dy/ds = y'/B shows that y' = 0. Hence the
principal normal is perpendicular to the generators of the cylinder. Since it is
also perpendicular to the tangent to the helix, it is normal to the cylinder, and
therefore the osculating plane is normal to the cylinder. It follows that the
SKEW CURVES [XI, §232
binormal lies in the tangent plane at right angles to the tangent to the helix ;
hence it also makes a constant angle with the z axis, i.e. y" is constant.
Since y' = 0, the formula dy'/ds = - y/B - y"/ T shows that y/R + y"/ T = 0;
hence the ratio T/R is constant for the helix.
Each of the properties mentioned above is characteristic for the helix. Let
us show, for example, that every curve for which the ratio T/R is constant is a
helix. (J. BERTRAND.)
From Frenet's formulae we have
dc^ _dp _dy _ T _ 1
da" ~ dp" ~ dy7' ~ R ~ H '
hence, if H is a constant, a single integration gives
a" = Ha - A , $" = Hp - B, y" = Hy - C ,
where A, B, C are three new constants. Adding these three equations after
multiplying them by a, /3, y, respectively, we find
Aa + Bp + Cy = H,
or
Aa + Bp + Cy H
\*A* + B* -r C2
But the three quantities
ABC
£2 + (72 ^/A2 + £2 + (72 Vvl2 + B2 + C2
are the direction cosines of a certain straight line A, and the preceding equa
tion shows that the tangent makes a constant angle with this line. Hence the
given curve is a helix.
Again, let us find the radius of curvature. By (53) and (61) we have
a' _da _ 1 pi
R ~ ~ds~ ~ -f
whence, since y' = 0,
<63) i
This shows that the ratio (1 + K2)/R is independent of K. But when K = 0
this ratio reduces to the reciprocal 1/r of the radius of curvature of the right
section C, which is easily verified (§ 205). Hence the preceding formula may
be written in the form R = r(l + K2), which shows that the ratio of the radius
of curvature of a helix to the radius of curvature of the corresponding curve C
is a constant.
It is now easy to find all the curves for which R and T are both constant.
For, since the ratio T/R is constant, all the curves must be helices, by Bertrand's
theorem. Moreover, since R is a constant, the radius of curvature r of the
curve C also is a constant. Hence C is a circle, and the required curve is a
helix which lies on a circular cylinder. This proposition is due to Puiseux.*
* It is assumed in this proof that we are dealing only with real curves, for we
assumed that A2 + B* + <?2 does not vanish. (See the thesis by Lyon : Sur les
courbes a torsion constante, 1890.)
XI, §233] CURVATURE TORSION 485
233. Bertrand's curves. The principal normals to a plane curve are also the
principal normals to an infinite number of other curves, — the parallels to the
given curve. J. Bertrand attempted to find in a similar manner all the skew
curves whose principal normals are the principal normals to a given skew
curve F. Let the coordinates x, y, z of a point of F be given as functions of the
arc ». Let us lay off on each principal normal a segment of length I, and let the
coordinates of the extremity of this segment be X, F, Z ; then we shall have
(64) X = x + la', Y = y + lp, Z = z + ly' .
The necessary and sufficient condition that the principal normal to the curve I"
described by the point (X, F, Z) should coincide with the principal normal to F
is that the two equations
a'dX + p'dY+y'dZ = 0,
y'(dXd2Y - dYd*X) = 0
should be satisfied simultaneously. The meaning of each of these equations is
evident. From the first, dl = 0 ; hence the length of the segment I should be a
constant. Replacing dX, d2 JT, dF, • • • in the second equation by their values
from Frenet's formulae and from the formulas obtained by differentiating
Frenet's, and then simplifying, we finally find
whence, integrating,
(65) I+l-1'.
where I' is the constant of integration. It follows that the required curves are
those for which there exists a linear relation between the curvature and the torsion.
On the other hand, it is easy to show that this condition is sufficient and that
the length I is given by the relation (05).
A remarkable particular case had already been solved by Monge, namely
that in which the radius of curvature is a constant. In that case (65) becomes
I = R, and the curve T" defined by the equations (64) is the locus of the centers
of curvature of F. From (64), assuming I — R = constant, we find the equations
which show that the tangent to T' is the polar line of F. The radius of curva
ture R' of F' is given by the formula
da"2 + d/3"2 + dy"*
hence R' also is constant and equal to R. The relation between the two curves
F and F' is therefore a reciprocal one : each of them is the edge of regression of
the polar surface of the other. It is easy to verify each of these statements for
the particular case of the circular helix.
486 SKEW CURVES [XI, §23*
Note. It is easy to find the general formulae for all skew curves whose radius of
curvature is constant. Let R be the given constant radius and let a, /3, 7 be any
three functions of a variable parameter which satisfy the relation a2 + /32 + y2 = 1.
Then the equations
(66) X =
where da- = Vda2 + d/32 + (fry2, represent a curve which has the required prop
erty, and it is easy to show that all curves which have that property may be
obtained in this manner. For a, /3, 7 are exactly the direction cosines of the
curve defined by (66), and <j is the arc of its spherical indicatrix (§ 225).
IV. CONTACT BETWEEN SKEW CURVES
CONTACT BETWEEN CURVES AND SURFACES
234. Contact between two curves. The order of contact of two
skew curves is defined in the same way as for plane curves. Let F
and I" be two curves which are tangent at a point A . To each point
M of F near A let us assign a point M' of F' according to such a law
that M and M' approach A simultaneously. We proceed to find
the maximum order of the infinitesimal MM' with respect to the
principal infinitesimal AM, the arc of F. If this maximum order
is n + 1, we shall say that the two curves have contact of order n.
Let us assume a system of trirectangular * axes in space, such
that the yz plane is not parallel to the common tangent at A, and
let the equations of the two curves be
( u— f(x\. ( Y = F(x),
/N •' ./ V / ' /fN \
If x0, y0, ZQ are the coordinates of A, the coordinates of M and M1
are, respectively,
[x0 -f h, f(x0 + A), <f>(x0 + /*)] , [arc + k, F(x0 + k}, 4>(.r0 + &)] ,
where A; is a function of h which is defined by the law of corre
spondence assumed between M and M' and which approaches zero
with h. We may select h as the principal infinitesimal instead of
the arc AM (§211); and a necessary condition that MM' should
be an infinitesimal of order n + 1 is that each of the differences
k-h, F(x() + 7c) - f(x0 + h} , &(x0 + fc) - <K*o + A)
* It is easy to show, by passing to the formula for the distance between two points
in oblique coordinates, that this assumption is not essential.
XI,§L»;a] CONTACT 487
should be an infinitesimal of order n + 1 or more. It follows that
we must have
k-h = ahn + l, F(x0 + K) -f(x0 + h) = /3hn + l,
<D(z0 + k) - <fr(x0 + h) = yhn + 1,
where a, ft, y remain finite as h approaches zero. Replacing k by
its value h + ahn+l from the first of these equations, the latter two
become
F(x, + h + «A» + 1) - /(*,, + A) = /8A- + 1,
$(x0 + A + <rA" + I) - <j>(X(> + A) = yA"*1.
Expanding F(x0 + h + ahn + l) and <t>(z0 + A + ahn+l) by Taylor's
series, all the terms which contain a will have a factor A" + * ; hence,
in order that the preceding condition be satisfied, each of the
differences
F(x0 + A) - f(xn + A) , 4>(a;0 + h) - <f>(X() + A)
should be of order n -f 1 or more. It follows that if MM' is of
order n + 1, the distance MN between the points M and N of the
two curves which have the same abscissa x0 -f h will be at least of
order n + 1. Hence the maximum order of the infinitesimal in
question will be obtained by putting into correspondence the points
of the two curves which have the same abscissa.
This maximum order is easily evaluated. Since the two curves
are tangent we shall have
Let us suppose for generality that we also have
but that at least one of the differences
does not vanish. Then the distance MM' will be of order n + 1
and the contact will be of order n. This result may also be stated
as follows : To find the order of contact of two curves Y and T', con
sider the two sets of projections (C, C") and (Cl, C{) of the given
curves on the xy plane and the xz plane, respectively, and find the
order of contact of each set ; then the order of contact of the given
curves F and I" will be the smaller of these two.
488 SKEW CURVES [XI, §236
If the two curves F and F' are given in the form
(F') X=.f(u), r=*(w), Z=*(M),
they will be tangent at a point u = t = t0 if
*(*o) = *('o) , *'('o) = *'(*o) , *(*o) = "KM , *'(*o) = <A'(V> •
T/* we suppose that f'(t0) is not zero, the tangent at the point of
contact is not parallel to the yz plane, and the points on the two
curves which have the same abscissa correspond to the same value
of t. In order that the contact should be of order n it is neces
sary and sufficient that each of the infinitesimals <$(£) — <£(Y) and
*(£) — \l/(t) should be of order n + 1 with respect to t — t0, i.e. that
we should have
*'(*o) = *'(*o) , — , *('° (*o) = 4>(n) Co) ,
*'(*«)= *'(<«), -, *(")(M = 'A(")(^),
and that at least one of the differences
should not vanish.
It is easy to reduce to the preceding the case in which one of the
curves F is given by equations of the form
(67) *
and the other curve F' by two implicit equations
Resuming the reasoning of § 212, we could show that a necessary
condition that the contact should be of order n at a point of F
where t = t0 is that we should have
(F(*.) = 0, F'(*o) = 0, -.., F<">(*0) = 0,
[Ft^-0, F|(«o) = 0, •••, Fi->(*0) = 0,
where
F(0 = nf(t)> *(0» -KO] , fr1 (0 = F,
235. Osculating curves. Let F be a curve whose equations are
given in the form (67), and let F' be one of a family of curves in
2n + 2 parameters a, b, c, • • •, I, which is defined by the equations
(69) F(x, y,z,a,b,-->,l) = 0, ^ (*, y, z,a,b,c, ••>, I) = 0.
XI, §235] CONTACT 489
In general it is possible to determine the 2n + 2 parameters in such
a way that the corresponding curve T' has contact of order rti with
the given curve F at a given point. The curve thus determined is
called the osculating curve of the family (69) to the curve T. The
equations which determine the values of the parameters a, b, c, • • • , I
are precisely the 2n + 2 equations (68). It should be noted that
these equations cannot be solved unless each of the functions F and
Fl contain at least n -f- 1 parameters. For example, if the curves
F' are plane curves, one of the equations (69) contains only three
parameters ; hence a plane curve cannot have contact of order
higher than two with a skew curve at a point taken at random on
the curve.
Let us apply this theory to the simpler classes of curves, — the
straight line and the circle. A straight line depends on four param
eters ; hence the osculating straight line will have contact of the
first order. It is easy to show that it coincides with the tangent,
for if we write the equations of the straight line in the form
x = az + p, y = bz + q,
the equations (68) become
where (x0, y0, 20) is the supposed point of contact on F. Solving
these equations, we find
which are precisely the values which give the tangent. A neces
sary condition that the tangent should have contact of the second
order is that x'J = az^' , y'0' = bztf, that is,
xo l/o zo
The points where this happens are those discussed in § 217.
The family of all circles in space depends on six parameters;
hence the osculating circle will have contact of the second order.
Let the equations of the circle be written in the form
F (x, y, z) = A(x - a) + B(y - b) + C(z - c~) =0,
F, (x, y, z) = (x- a)2 + (y - W + (« - c)2 - /22 = 0,
490 SKEW CURVES [XI, §236
where the parameters are a, b, c, R, and the two ratios of the three
coefficients A, B, C. The equations which determine the osculating
circle are
A(x - «) -f B(y -b} + C(z — e) = 0,
A*2+B%+C*1-0,
at at at
(x - a)2 + (y -
C— -H>-
where x, y, and s are to be replaced by /(£), <£(£), and ^(£), respec
tively. The second and the third of these equations show that the
plane of the osculating circle is the osculating plane of the curve F.
If a, b, and c be thought of as the running coordinates, the last
two equations represent, respectively, the normal plane at the point
(x, y, z) and the normal plane at a point whose distance from
(x, y, z) is infinitesimal. Hence the center of the osculating circle
is the point of intersection of the osculating plane and the polar
line. It follows that the osculating circle coincides with the circle
of curvature, as we might have foreseen by noticing that two curves
which have contact of the second order have the same circle of
curvature, since the values of y1, z', y", z" are the same for the two
curves.
236. Contact between a curve and a surface. Let S be a surface
and T a curve tangent to S at a point A. To any point M of T
near A let us assign a point M' of 5 according to such a law that
M and M ' approach A simultaneously. First let us try to find what
law of correspondence between M and M' will render the order
of the infinitesimal MM' with respect to the arc AM a. maximum.
Let us choose a system of rectangular coordinates in such a way
that the tangent to T shall not be parallel to the yz plane, and that
the tangent plane to S shall not be parallel to the z axis. Let
(*oi y<» *o) be the coordinates of A ; Z = F(x, y) the equation of S ;
y =f(x), z = <fr(x) the equations of T ; and n + 1 the order of the
infinitesimal MM' for the given law of correspondence. The
CONTACT 491
coordinates of M are [x0 + h, f(x0 + A), <f>(x0 + &)]. Let X, Y, and
Z = F(X, Y) be the coordinates of M'. In order that MM' should
be of order n + 1 with respect to the arc AM, or, what amounts to
the same thing, with respect to h, it is necessary that each of the
differences X — - x, Y — y, and Z — z should be an infinitesimal at
least of order n + 1, that is, that we should have
X-x = ah" + 1, Y-y = /3hn + l, Z - z = F(X, F) - z = yhn + l,
where a, ft, y remain finite as h approaches zero. Hence we shall
have
F(x + ahn + l, y + (3hn + l) -z = yhn + l,
and the difference F(x, y) — z will be itself at least of order n + 1.
This shows that the order of the infinitesimal MN, where N is the
point where a parallel to the z axis pierces the surface, will be at
least as great as that of MM'. The maximum order of contact —
which we shall call the order of contact of the curve and the surface
— is therefore that of the distance MN with respect to the arc AM
or with respect to h. Or, again, we may say that the order of con
tact of the curve and the surface is the order of contact between T
and the curve T' in which the surface S is cut by the cylinder which
projects T upon the xy plane. (It is evident that the z axis may be
any line not parallel to the tangent plane.) For the equations of
the curve T' are
y=f(x), Z=F[>, /(*)] = *(*),
and, by hypothesis,
*(*„) = «K*o), *'(*„) = *'(*„).
If we also have
the curve and the surface have contact of order n. Since the equa
tion $(x) = <t>(x) gives the abscissae of the points of intersection of
the curve and the surface, these conditions for contact of order n
at a point A may be expressed by saying that the curve meets the
surface in n -f 1 coincident points at A.
Finally, if the curve T is given by equations of the form x =f(f),
y = <£(<), z = \fr(f), and the surface S is given by a singly equation
of the form F(x, y, z} = 0, the curve T' just defined will have equa
tions of the form x =/(*), y = <£(£), * = w(£), where ir(t) is a func
tion defined by the equation
» +(«),*•(*)] »0.
492 SKEW CURVES [XI, §237
In order that F and F' should have contact of order n, the infini
tesimal 7r(Y) — \l/(f) must be of order n + 1 with respect to t — t0 ;
that is, we must have
Using F(£) to denote the function considered in § 234, these equa
tions may be written in the form
These conditions may be expressed by saying that the curve and
the surface have n + 1 coincident points of intersection at their
point of contact.
If 5 be one of a family of surfaces which depends on n + 1
parameters a, b, c, •••, I, the parameters may be so chosen that S
has contact of order n with a given curve at a given point ; this
surface is called the osculating surface.
In the case of a plane there are three parameters. The equations
which determine these parameters for the osculating plane are
Af (t} + B<j> (t) + C$ (t) + D = 0,
Af (t) + B# (t) + Cy (£) = 0,
Af"(t) + B4"(t) + C^"(f) = 0.
It is clear that these are the same equations we found before for
the osculating plane, and that the contact is in general of the second
order. If the order of contact is higher, we must have
Af'"(t) + Bt"'(£) + Cf"(f) = 0,
i.e. the osculating plane must be stationary.
237. Osculating sphere. The equation of a sphere depends on four
parameters ; hence the osculating sphere will have contact of the
third order. For simplicity let us suppose that the coordinates
x, y, K of a point of the given curve F are expressed in terms of the
arc s of that curve. In order that a sphere whose center is (a, b, c}
and whose radius is p should have contact of the third order with
F at a given point (a;, y, z) on F, we must have
F(*) = 0, F'(*) = 0, F"(*) = 0, F"(*) = 0,
where
F(«) = (x - a)2 -f- (y - i)2 + (z - c)2 - p*
XI, §238] CONTACT 493
and where x, y, z are expressed as functions of s. Expanding the
last three of the equations of condition and applying Frenet's
formulae, we find
F' 0) = (x - a) a + (y - b) {3 + (* - c)y = 0,
' O/ - *) + (* - ') +1= 0,
/z \R T R
These three equations determine a, 6, and c. But the first of them
represents the normal plane to the curve F at the point (x, y, z) in
the running coordinates (a, b, c), and the other two may be derived
from this one by differentiating twice with respect to s. Hence
the center of the osculating sphere is the point where the polar line
touches its envelope. In order to solve the three equations we may
reduce the last one by means of the others to the form
J D
(x - a)a» + (y - J)0r, + (, _ c)yr = T —,
from which it is easy to derive the formulae
a = x + Ra'-T^a", b = y + R? - T ~
Hence the radius of the osculating sphere is given by the formula
If R is constant, the center of the osculating sphere coincides with
the center of curvature, which agrees with the result obtained in
§233.
238. Osculating straight lines. If the equations of a family of
curves depend on n -f 2 parameters, the parameters may be chosen
in such a way that the resulting curve C has contact of order n with
a given surface S at a point M. For the equation which expresses
that C meets S at M and the n -f 1 equations which express that
there are n -f- 1 coincident points of intersection at M constitute
n + 2 equations for the determination of the parameters.
494 SKEW CURVES [XI, EM.
For example, the equations of a straight line depend on four
parameters. Hence, through each point M of a given surface S,
there exist one or more straight lines which have contact of the
second order with the surface. In order to determine these lines,
let us take the origin at the point M, and let us suppose that the
z axis is not parallel to the tangent plane at M. Let z = F(x, y)
be the equation of the surface with respect to these axes. The
required line evidently passes through the origin, and its equations
are of the form
x _ y _ z _
a b c
Hence the equation cp = F(ap, bp) should have a triple root p = 0 ;
that is, we should have
c = ap + bq,
where p, q, r, s, t denote the values of the first and second deriva
tives of F(x, y) at the origin. The first of these equations expresses
that the required line lies in the tangent plane, which is evident
a priori. The second equation is a quadratic equation in the ratio
ft/a, and its roots are real if s2 — rt is positive. Hence there are in
general two and only two straight lines through any point of a given
surface which have contact of the second order with that surface.
These lines will be real or imaginary according as s2 — rt is positive
or negative. We shall meet these lines again in the following
chapter, in the study of the curvature of surfaces.
EXERCISES
1. Find, in finite form, the equations of the evolutes of the curve which
cuts the straight line generators of a right circular cone at a constant angle.
Discuss the problem.
[Licence, Marseilles, July, 1884.]
2. Do there exist skew curves T for which the three points of intersection
of a fixed plane P with the tangent, the principal normal, and the binormal are
the vertices of an equilateral triangle ?
3. Let T be the edge of regression of a surface which is the envelope of
a one-parameter family of spheres, i.e. the envelope of the characteristic circles.
Show that the curve which is the locus of the centers of the spheres lies on
the polar surface of T. Also state and prove the converse.
4. Let T be a given skew curve, M a point on T, and 0 a fixed point in
space. Through 0 draw a line parallel to the polar line to T at M, and lay off
on this parallel a segment ON equal to the radius of curvature of F at M. Show
XI, Exs.] EXERCISES 495
that the curve F' described by the point N and the curve T" described by the
center of curvature of F have their tangents perpendicular, their elements of
length equal, and their radii of curvature equal, at corresponding points.
[ROUQUET.]
5. If the osculating sphere to a given skew curve F has a constant radius a,
show that F lies on a sphere of radius a, at least unless the radius of curvature
of F is constant and equal to a.
6. Show that the necessary and sufficient condition that the locus of the
center of curvature of a helix drawn on a cylinder should be another helix on a
cylinder parallel to the first one is that the right section of the second cylinder
should be a circle or a logarithmic spiral. In the latter case show that all the
helices lie on circular cones which have the same axis and the same vertex.
[Tissox, Nouvelles Annales, Vol. XI, 1852.]
7*. If two skew curves have the same principal normals, the osculating
planes of the two curves at the points where they meet the same normal make
a constant angle with each other. The two points just mentioned and the cen
ters of curvature of the two curves form a system of four points whose anhar-
monic ratio is constant. The product of the radii of torsion of the two curves
at corresponding points is a constant.
[PAUL SERRET ; MANNHEIM ; SCHELL.]
8*. Let x, y, z be the rectangular coordinates of a point on a skew curve F,
and s the arc of that curve. Then the curve F0 defined by the equations
X0 = I a"ds, yo — I P"ds, z0 = ty"ds,
where x0, yo, z<> are the running coordinates, is called the conjugate curve to F;
and the curve defined by the equations
Z0sin0, T= y cos0 + y0sin0, Z = z cos0 -f 20sin0,
where JT, F, Z are the running coordinates and 6 is a constant angle, is called
a related curve. Find the orientation of the fundamental trihedron for each of
these curves, and find their radii of curvature and of torsion.
If the curvature of F is constant, the torsion of the curve F0 is constant, and
the related curves are curves of the Bertrand type (§ 233). Hence find the
general equations of the latter curves.
9. Let F and I" be two skew curves which are tangent at a point A. From
A lay off infinitesimal arcs AM and AM' from A along the two curves in the
same direction. Find the limiting position of the line MM'.
[CAUGHT.]
10. In order that a straight line rigidly connected to the fundamental trihe
dron of a skew curve and passing through the vertex of the trihedron should
describe a developable surface, that straight line must coincide with the tangent,
at least unless the given skew curve is a helix. In the latter case there are an
infinite number of straight lines which have the required property.
496 SKEW CURVES [XI, Exs.
For a curve of the Bertrand type there exist two hyperbolic paraboloids
rigidly connected to the fundamental trihedron, each of whose generators
describes a developable surface.
, Bivista di Mathematical,, Vol. II, 1892, p. 155.]
11*. In order that the principal normals of a given skew curve should be the
binormals of another curve, the radii of curvature and the radii of torsion of
the first curve must satisfy a relation of the form
A
where A and B are constants.
/JL , 1\ =
\R* TV
[MANNHEIM, Comptes rendus, 1877.]
[The case in which a straight line through a point on a skew curve rigidly
connected with the fundamental trihedron is also the principal normal (or the
binormal) of another skew curve has been discussed by Pellet (Comptes rendus,
May, 1887), by Cesaro (Nouvelles Annales, 1888, p. 147), and by Balitrand
(Mathesis, 1894, p. 159).]
12. If the osculating plane to a skew curve F is always tangent to a fixed
sphere whose center is 0, show that the plane through the tangent perpen
dicular to the principal normal passes through 0, and show that the ratio of
the radius of curvature to the radius of torsion is a linear function of the arc.
State and prove the converse theorems.
CHAPTER XII
SURFACES
I. CURVATURE OF CURVES DRAWN ON A SURFACE
239. Fundamental formula. Meusnier's theorem. In order to study
the curvature of a surface at a non-singular point M, we shall sup
pose the surface referred to a system of rectangular coordinates
such that the axis of z is not parallel to the tangent plane at M.
If the surface is analytic, its equation may be written in the form
(1) * = F(x,y),
where F(x, ?/) is developable in power series according to powers of
x — XQ and y — y0 in the neighborhood of the point M (x0, y0, «0)
(§ 194). But the arguments which we shall use do not require the
assumption that the surface should be analytic : we shall merely
suppose that the function F(x, ?/), together with its first and second
derivatives, is continuous near the point (x0, y0~) We shall use
Monge's notation, p, q, r, s, t, for these derivatives.
It is seen immediately from the equation of the tangent plane
that the direction, cosines of the normal to the surface are propor
tional to p, q, and — 1. If we adopt as the positive direction of the
normal that which makes an acute angle with the positive z axis,
the actual direction cosines themselves A, /*, v are given by the
formulae
\ -
A —
u, =
Let C be a curve on the surface S through the point M, and let
the equations of this curve be given in parameter form ; then the
functions of the parameter which represent the coordinates of a
point of this curve satisfy the equation (1), and hence their differ
entials satisfy the two relations
(3) dz = p dx + q di/ ,
(4) <Pz = p d*x + q d*y + r dx2 + 2s dxdy + t dy*.
497
498 SURFACES [XII, §239
*
The first of these equations means that the tangent to the curve C
lies in the tangent plane to the surface. In order to interpret the
second geometrically, let us express the differentials which occur in
it in terms of known geometrical quantities. If the independent
variable be the arc a of the curve C, we shall have
dx:_ ^/_o ^f_ d*x_<^ &y__P_ <&* = y'
~fo ~ a' do- ~ "' do-~y' dv* ~ R ' da2 ~ R ' da2 ~ R '
where the letters a, ft, y, a', /3', y', R have the same meanings as in
§ 229. Substituting these values in (4) and dividing by
that equation becomes
y -pa'-gp* =
R^/l + p* + q2
or, by (2),
\a' + fji/3' + vy' = ra2 + 2sa@ +
R
But the numerator Xa' + p.(3' + vy' is nothing but the cosine of the
angle 0 included between the principal normal to C and the positive
direction of the normal to the surface ; hence the preceding formula
may be written in the form
COS 6 ra2 + 2sa/3 + tfB*
(5)
R
This formula is exactly equivalent to the formula (4); hence it
contains all the information we can discover concerning the curva
ture of curves drawn on the surface. Since R and Vl + p* + q2
are both essentially positive, cos 0 and ra2 -f 2saft + tf}2 have the same
sign, i.e. the sign of the latter quantity shows whether 0 is acute or
obtuse. In the first place, let us consider all the curves on the sur
face S through the point M which have the same osculating plane
(which shall be other than the tangent plane) at the point M. All
these curves have the same tangent, namely the intersection of the
osculating plane with the tangent plane to the surface. The direc
tion cosines a, /?, y therefore coincide for all these curves. Again,
the principal normal to any of these curves coincides with one of
the two directions which can be selected upon the perpendicular to the
tangent line in the osculating plane. Let o> be the angle which the
normal to the surface makes with one of these directions ; then we
shall have 6 = o> or 0 = TT — to. But the sign of ra'2 + 2sa/3 + tfP
shows whether the angle 0 is acute or obtuse ; hence the positive
XII, §239] CURVES ON A SURFACE 499
direction of the principal normal is the same for all these curves.
Since 6 is also the same for all the curves, the radius of curvature
R is the same for them all ; that is to say, all the curves on the sur
face through the point M which have the same osculating plane have
the same center of curvature.
It follows that we need only study the curvature of the plane
sections of the surface. First let us study the variation of the
curvature of the sections of the surface by planes which all pass
through the same tangent MT. We may suppose, without loss of
generality, that raz + 2sa(3 + fy32 > 0, for a change in the direction
of the z axis is sufficient to change the signs of r, s, and t. For all
these plane sections we shall have, therefore, cos 0 > 0, and the
angle 6 is acute. If R! be the radius of curvature of the section
by the normal plane through MT, since the corresponding angle 0
is zero, we shall have
1 __ ra* + 2sa(3 + t/32
Rl
Comparing this formula with equation (5), which gives the radius
of curvature of any oblique section, we find
1 cos0
W^-JT'
or R = R! cos 0, which shows that the center of curvature of any
oblique section is the projection of the center of curvature of the
normal section through the same tangent line. This is Meusnier's
theorem.
The preceding theorem reduces the study of the curvature of
oblique sections to the study of the curvature of normal sections.
We shall discuss directly the results obtained by Euler. First let
us remark that the formula (5) will appear in two different forms
for a normal section according as ra2 + 2saft + t{P is positive or
negative. In order to avoid the inconvenience of carrying these
two signs, we shall agree to affix the sign + or the sign — to the
radius of curvature R of a normal section according as the direction
from M to the center of curvature of the section is the same as or
opposite to the positive direction of the normal to the surface.
With this convention, R is given in either case by the formula
(7) 1 = ra* + 28ap + tF
R
500 SURFACES [XII, §239
which shows without ambiguity the direction in which the center
of curvature lies.
From (7) it is easy to determine the position of the surface with
respect to its tangent plane near the point of tangency. For if
s2 — rt < 0, the quadratic form ra2 + 2saft •}- tfi2 keeps the same
sign — the sign of r and of t — as the normal plane turns around
the normal; hence all the normal sections have their centers of
curvature on the same side of the tangent plane, and therefore all
lie on the same side of that plane : the surface is said to be convex
at such a point, and the point is called an elliptic point. On the
contrary, if s2 — rt > 0, the form m2 + 2sa(3 + tfi2 vanishes for two
particular positions of the normal plane, and the corresponding
normal sections have, in general, a point of inflection. When the
normal plane lies in one of the dihedral angles formed by these two
planes, R is positive, and the corresponding section lies above the tan
gent plane ; when the normal plane lies in the other dihedral angle,
R is negative, and the section lies below the tangent plane. Hence
in this case the surface crosses its tangent plane at the point of
tangency. Such a point is called a hyperbolic point. Finally, if
s2 — rt = 0, all the normal sections lie on the same side of the tan
gent plane near the point of tangency except that one for which
the radius of curvature is infinite. The latter section usually
crosses the tangent plane. Such a point is called a parabolic point.
It is easy to verify these results by a direct study of the differ
ence n = z — z' of the values of z for a point on the surface and for
the point on the tangent plane at M which projects into the same
point (x, y} on the xy plane. For we have
z' = p(x — a*0) + q(ij — i/0) ,
whence, for the point of tangency (xn, ?/0),
du _ cz' _ ^M_n
dx dx dy
and
d2 u (ft u d2 u
It follows that if s2 — rt < 0, u is a maximum or a minimum at M
(§ 56), and since u vanishes at M. it has the same sign for all other
points in the neighborhood. On the other hand, if s2 — rt > 0, u
has neither a maximum nor a minimum at M, and hence it changes
sign in any neighborhood of M.
XII, §240] CURVES ON A SURFACE 501
240. Euler's theorems. The indicatrix. In order to study the varia
tion of the radius of curvature of a normal section, let us take the
point M as the origin and the tangent plane at M as the xy plane.
With such a system of axes we shall have p = q = 0, and the
formula (7) becomes
(8) — = r cos2<£ + 2s cos <f> sin <£ + t sin2<£,
where <£ is the angle which the trace of the normal plane makes
with the positive x axis. Equating the derivative of the second
member to zero, we find that the points at which R may be a maxi
mum or a minimum stand at right angles. The following geomet
rical picture is a convenient means of visualizing the variation of R.
Let us lay off, on the line of intersection of the normal plane with
the xy plane, from the origin, a length Om equal numerically to the
square root of the absolute value of the corresponding radius of cur
vature. The point ra will describe a curve, which gives an instanta
neous picture of the variation of the radius of curvature. This ciirve
is called the indicatrix. Let us examine the three possible cases.
1) s2 — rt < 0. In this case the radius R has a constant sign, which
we shall suppose positive. The coordinates of m are £ = V# cos <£
and r) — V-R sin <£ ; hence the equation of the indicatrix is
(9) re
which is the equation of an ellipse whose center is the origin. It is
clear that R is at a maximum for the section made by the normal
plane through the major axis of this ellipse, and at a minimum for
the normal plane through the minor axis. The sections made by two
planes which are equally inclined to the two axes evidently have the
same curvature. The two sections whose planes pass through the
axes of the indicatrix are called the principal normal sections, and
the corresponding radii of curvature are called the principal radii of
curvature. If the axes of the indicatrix are taken for the axes of x
and y, we shall have s — 0, and the formula (8) becomes
— = T cos2<£ + t sin2<£.
R
With these axes the principal radii of curvature Rl and R2 correspond
to <j> = 0 and <f> — Tr/2, respectively ; hence 1/Ri = r, 1/R2 = t, and
(10) 1 ^cos2<ft > sin2<fr
R R R
502 SURFACES [XII, § 240
2) s2 — rt > 0. The normal sections which correspond to the
values of <f> which satisfy the equation
r cos2<£ + 2s cos <j> sin <£ + t sin2<£ = 0
have infinite radii of curvature. Let L(OL^ and L^OL2 be the inter
sections of these two planes with the xy plane. When the trace of
the normal plane lies in the angle L^OL^^ for example, the radius
of curvature is positive. Hence the corresponding portion of the
indicatrix is represented by the equation
where £ and 77 are, as in the previous case, the coordinates of the
point m. This is an hyperbola whose asymptotes are the lines
L[OLl and L'ZOLZ. When the trace of the normal plane lies in the
other angle L'^OLlt R is negative, and the coordinates of m are
£ = V— R cos <f), t] — V— R sin <£.
Hence the corresponding portion of the indicatrix is the hyperbola
which is conjugate to the preceding hyperbola. These two hyper
bolas together form a picture of the variation of the radius of curva
ture in this case. If the axes of the hyperbolas be taken as the
x and y axes, the formula (8) may be written in the form (10), as in
the previous case, where now, however, the principal radii of curva
ture R± and R2 have opposite signs.
3) s2 — rt = 0. In this case the radius of curvature R has a
fixed sign, which we shall suppose positive. The indicatrix is still
represented by the equation (9), but, since its center is at the origin
and it is of the parabolic type, it must be composed of two parallel
straight lines. If the axis of y be taken parallel to these lines, we
shall have s = 0, t — 0, and the general formula (8) becomes
- = rcos*$,
it
or
R
This case may also be considered to be a limiting case of either of
the preceding, and the formula just found may be thought of as the
limiting case of (10), when R2 becomes infinite.
XII, §241] CURVES ON A SURFACE 503
Euler'a formulae may be established without using the formula (5). Taking
the point M of the given surface as the origin and the tangent plane as the xy
plane, the expansion of z by Taylor's series may be written in the form
rz2 + 2sxy + ty*
~o- - + -"'
where the terms not written down are of order greater than two. In order
to find the radii of curvature of the section made by a plane y = x tan 0, we
may introduce the transformation
x = x' cos 0 — y' sin <t> , y = x' sin 0 + y' cos <£ ,
and then set y' = 0. This gives the expansion of z in powers of x',
_ r cos2 <(> + 2s sin <f> cos <f> + t sin2 0
2 — 2J — |- • • •
1.2
which, by § 214, leads to the formula (8).
Notes. The section of the surface by its tangent plane is given by the equation
0 = rz2 + 2sxy + ty* + <j>3 (x, y) + • • • ,
and has a double point at the origin. The two tangents at this point are the
asymptotic tangents. More generally, if two surfaces S and Si are both tangent
at the origin to the xy plane, the projection of their curve of intersection on the
xy plane is given by the equation
0 = (r - n)x2 + 2(8 - Sl)xy + (t - t^y* + • • -,
where r\, s\, ti have the same meaning for the surface Si that r, s, t have
for S. The nature of the double point depends upon the sign of the expression
(s — Si)2 — (r — ri)(t — ti). If this expression is zero, the curve of intersection
has, in general, a cusp at the origin.
To recapitulate, there exist on any surface four remarkable posi
tions for the tangent at any point : two perpendicular tangents for
which the corresponding radii of curvature have a maximum or a
minimum, and two so-called asymptotic, or principal,* tangents, for
which the corresponding radii of curvature are infinite. The latter are
to be found by equating the trinomial ra2-+-2saft + t{P to zero (§ 238).
We proceed to show how to find the principal normal sections and
the principal radii of curvature for any system of rectangular axes.
241. Principal radii of curvature. There are in general two different
normal sections whose radii of curvature are equal to any given
value of R. The only exception is the case in which the given
value of R is one of the principal radii of curvature, in which case
* The reader should distinguish sharply the directions of the principal tangents
(the asymptotes of the indicatrix) and the directions of the principal normal sections
(the axes of the indicatrix) . To avoid confusion we shall not use the term principal
tangent. — TRANS.
504 SURFACES
[XII, § 241
only the corresponding principal section has the assigned radius
of curvature. To determine the normal sections whose radius of
curvature is a given number R, we may determine the values of
a, ft, y by the three equations
Vl + p2 + a*
-J— = ra* + 2sa/3 + tft\ y=pa + qft, a2 + /32 + y2=l.
It is easy to derive from these the following homogeneous combina
tion of degree zero in a and ft :
R <** + ft* + (pa + qpf
It follows that the ratio ft/ a is given by the equation
a\l + p* - rD) + 2aft(pq - sD) + (?(!+ <f - tD) = 0,
where R — D Vl +p2 + q2. If this equation has a double root, that
root satisfies each of the equations formed by setting the two first
derivatives of the left-hand side with respect to a and ft equal to
zero :
SD) =0,
(12)
I a(pq- sD} + ft(l + q* - tD) = 0.
Eliminating a and £ and replacing D by its value, we obtain an
equation for the principal radii of curvature :
On the other hand, eliminating D from the equations (12), we obtain
an equation of the second degree which determines the lines of inter
section of the tangent plane with the principal normal sections :
-pqr]
From the very nature of the problem the roots of the equations (13)
and (14) will surely be real. It is easy to verify this fact directly.
In order that the equation for R should have equal roots, it is
necessary that the indicatrix should be a circle, in which case all
the normal sections will have the same radius of curvature. Hence
the second member of (11) must be independent of the ratio ft fa,
which necessitates the equations
pq
XII, §241] CURVES ON A SURFACE 505
The points which satisfy these equations are called umbilics. At
such points the equation (14) reduces to an identity, since every
diameter of a circle is also an axis of symmetry.
It is often possible to determine th'e principal normal sections
from certain geometrical considerations. For instance, if a surface
S has a plane of symmetry through a point M on the surface, it is
clear that the line of intersection of that plane with the tangent
plane at M is a line of symmetry of the indicatrix ; hence the sec
tion by the plane of symmetry is one of the principal sections. For
example, on a surface of revolution the meridian through any point
is one of the principal normal sections ; it is evident that the plane
of the other principal normal section passes through the normal to
the surface and the tangent to the circular parallel at the point.
But we know the center of curvature of one of the oblique sections
through this tangent line, namely that of the circular parallel itself.
It follows from Meusnier's theorem that the center of curvature of
the second principal section is the point where the normal to the
surface meets the axis of revolution.
At any point of a developable surface, s2 — rt = 0, and the indica
trix is a pair of parallel straight lines. One of the principal sec
tions coincides with the generator, and the corresponding radius of
curvature is infinite. The plane of the second principal section is
perpendicular to the generator. All the points of a developable
surface are parabolic, and, conversely, these are the only surfaces
which have that property (§ 222).
If a non-developable surface is convex at certain points, while other
points of the surface are hyperbolic, there is usually a line of para
bolic points which separates the region where s2 — rt is positive from
the region where the same quantity is negative. For example, on the
anchor ring, these parabolic lines are the extreme circular parallels.
In general there are on any convex surface only a finite number of umbilics.
We proceed to show that the only real surface for which every point is an
umbilic is the sphere. Let X, p, v be the direction cosines of the normal to the
surface. Differentiating (2), we find the formulae «
ax _ pqs-(l+q*)r 5X _ pqt-(l + q*)s
CX ~ (l + p2+92)J ' Sy
dfj. _ pqr-(l+p*)s dp
or, by (15),
— = 0 ^ = 0 — = 8--
dy ~ ' dx ~ ' dx ~ dy '
500 SURFACES [XII, § 242
The first equation shows that X is independent of y, the second that /x is inde
pendent of z ; hence the common value of d\/cx, dp/dy is independent of both
x and y, i.e. it is a constant, say I/a. This fact leads to the equations
x - X0 y -Mo Va2 - (x - x0)2 - (y - 2/o)2
X= — - — , n
a
X — X0
Va2 - (x - x0)2 - (y - yo)2
y -2/0 )
Va2-(x-x0)2-(2/-2/o)2'
whence, integrating, the value of z is found to be
z = z0 + Va2 - (x - x0)2 - (y - y0)* ,
which is the equation of a sphere. It is evident that if 8\/dx = dp/dy = 0, the
surface is a plane. But the equations (15) also have an infinite number of
imaginary solutions which satisfy the relation 1 + p2 + q2 = 0, as we can see by
differentiating this equation with respect to x and with respect to y.
II. ASYMPTOTIC LINES CONJUGATE LINES
242. Definition and properties of asymptotic lines. At every hyper
bolic point of a surface there are two tangents for which the corre
sponding normal sections have infinite radii of curvature, namely
the asymptotes of the indicatrix. The curves on the given surface
which are tangent at each of their points to one of these asymptotic
directions are called asymptotic lines. If a point moves along any
curve on a surface, the differentials dx, dy, dz are proportional to
the direction cosines of the tangent. For an asymptotic tangent
roP + 2saf3 + tft* = 0 ; hence the differentials dx and dy at any point
of an asymptotic line must satisfy the relation
(16) rdx2 + 2sdxdy + tdif = 0.
If the equation of the surface be given in the form z = F(x, y), and
we substitute for r, s, and t their values as functions of x and y,
this equation may be solved for dy/dx, and we shall obtain the two
solutions
We shall see later that each of these equations has an infinite num
ber of solutions, and that every pair of values (x0, y0) determines
in general one and only one solution. It follows that there pass
through every point of the surface, in general, two and only two
XII, §242] ASYMPTOTIC LINES CONJUGATE LINES 507
asymptotic lines : all these lines together form a double system of
lines upon the surface.
Again, the asymptotic lines may be defined without the use of
any metrical relation : the asymptotic lines on a surface are those
curves for which the osculating plane always coincides with the tan
gent plane to the surface. For the necessary and sufficient condition
that the osculating plane should coincide with the tangent plane to
the surface is that the equations
dz — p dx — q dy = 0 , d2z — p d^x — q d*y = 0
should be satisfied simultaneously (see § 215). The first of these
equations is satisfied by any curve which lies on the surface. Dif
ferentiating it, we obtain the equation
d*z — p d*x — q d*y — dp dx — dq dy = 0 ,
which shows that the second of the preceding equations may be
replaced by the following relation between the first differentials :
(18) dp dx + dqdy= 0,
an equation which coincides with (16). Moreover it is easy to
explain why the two definitions are equivalent. Since the radius of
curvature of the normal section which is tangent to an asymptote
of the indicatrix is infinite, the radius of curvature of the asymp
totic line will also be infinite, by -Meusnier's theorem, at least unless
the osculating plane is perpendicular to the normal plane, in which
case Meusnier's theorem becomes illusory. Hence the osculating
plane to an asymptotic line must coincide with the tangent plane,
at least unless the radius of curvature is infinite ; but if this were
true, the line would be a straight line and its osculating plane
would be indeterminate. It follows from this property that any
projective transformation carries the asymptotic lines into asymp
totic lines. It is evident also that the differential equation is of
the same form whether the axes are rectangular or oblique, for the
equation of the osculating plane remains of the same form.
It is clear that the asymptotic lines exist only in case the points of
the surface are hyperbolic. But when the surface is analytic the
differential equation (16) always has an infinite number of solu
tions, real or imaginary, whether s1 — rt is positive or negative. As a
generalization we shall say that any convex surface possesses two sys
tems of imaginary asymptotic lines. Thus the asymptotic lines of an
unparted hyperboloid are the two systems of rectilinear generators.
508 SURFACES [XII, §243
For an ellipsoid or a sphere these generators are imaginary, but
they satisfy the differential equation for the asymptotic lines.
Example. Let us try to find the asymptotic lines of the surface
z = xmy*.
In this example we have
r = m(m — l)xm-2yn, s — mnxm-lyn-1, t = n(n — l)xmy"-2,
and the differential equation (16) may be written in the form
x dy/ \x dy
This equation may be solved as a quadratic in (ydx)/(xdy). Let hi and h^ be
the solutions. Then the two families of asymptotic lines are the curves which
project, on the xy plane, into the curves
243. Differential equation in parameter form. Let the equations of
the surface be given in terms of two parameters u and v :
(19) x=f(u, v), y = $(u,v}, z = ^(u,v}.
Using the second definition of asymptotic lines, let us write the
equation of the tangent plane in the form
(20) A(X - x) + B(Y - y) + C(Z - z) = 0,
where A, B, and C satisfy the equations
-
dv ov 8v
which are the equations for A, B, and C found in § 39. Since the
osculating plane of an asymptotic line is the same as this tangent
plane, these same coefficients must satisfy the equations
Adx + Bdij + Cdz =0,
Ad*x + Bd*y + Cd?z = 0.
The first of these equations, as above, is satisfied identically. Differ
entiating it, we see that the second may be replaced by the equation
(22) dA dx + dBdy + dCdz = 0,
which is the required differential equation. If, for example, we
set C = — 1 in the equations (21), A and B are equal, respectively,
to the partial derivatives p and q of z with respect to x and y, and
the equation (22) coincides with (18).
XII, §244] ASYMPTOTIC LINES CONJUGATE LINES
509
Examples. As an example let us consider the conoid z = <f>(y/x). This equa
tion is equivalent to the system x = u, y = w>, z = </>(«), and the equations (21)
become
A + Bv = 0 , Bu + C<j>'(v) - 0 .
These equations are satisfied if we set C = — u, A = — «#'(«), .B = #'(») ; hence
the equation (22) takes the form
utf>"(v)dv'2 — 24>'(v)dudv = 0.
One solution of this equation is v = const., which gives the rectilinear genera
tors. Dividing by dv, the remaining equation is
<t>"(v) dv _ 2 du
<j>'(v) u
whence the second system of asymptotic lines are the curves on the surface
defined by the equation w2 = K<f>' '(v) , which project on the xy plane into the
curves
Again, consider the surfaces discussed by Jamet, whose equation may be
written in the form
Taking the independent variables z and u = y/x, the differential equation of
the asymptotic lines may be written in the form
JTW \ /(u>
from which each of the systems of asymptotic lines may be found by a single
quadrature.
A helicoid is a surface defined by equations of the form
The reader may show that the differential equation of the asymptotic lines is
pf"(p) dp* - 2h dw dp + p*f'(p) du* = 0 ,
from which w may be found by a single quadrature.
244. Asymptotic lines on a ruled surface. Eliminating A , B, and C
between the equations (21) and the equation
A /-/2 n* I D sj* .-I* I /"* ^-72 fj _ _ ||
JT. \JU *J ^^ J-f U/ I/ [^ \*> Uv & — — V/ y
we find the general differential equation of the asymptotic lines :
d f d <j> diff
du du du
(23) df 8$ d$ =0.
dv dv dv
510
SURFACES
[XII, § 244
This equation does not contain the second differentials dzu and d2v,
for we have
CV
CUCV
du dv
ov*
dv*
and analogous expressions for d2y and d2z. Subtracting from the
third row of the determinant (23) the first row multiplied by d2u
and the second row multiplied by d2v, the differential equation
becomes
cu
df_
_ _
du en
c_$ cj,
do cv
cu cv
vtr
= 0.
Developing this determinant with respect to the elements of the
first row and arranging with respect to du and dv, the equation
may be written in the form
(24) D du2 + 2D' du dv + D" dv* = 0,
where D, D', and D" denote the three determinants
(25)
dx dy dz
dll du du
dx dy dz
cu du du
»•-
dx dy dz
dv dv cv
, D' =
dx dy dz
dv dv dv
d2x d2y d2z
c2x d2y d2z
du2 du2 du2
du dv du dv du dv
dx dy dz
du du cu
, »"=
dx dy dz
dv dv dv
•
fo2 'dv2 W2
As an application let us consider a ruled surface, that is, a surface
whose equations are of the form
where x0, y0, z0, a, /8, y are all functions of a second variable param
eter v. If we set u = 0, the point (a-0, y0, z0) describes a certain
curve F which lies on the surface. On the other hand, if we set
v — const, and let u vary, the point (x, y, z) will describe a straight-
XII, §245] ASYMPTOTIC LINES CONJUGATE LINES 511
line generator of the ruled surface, and the value of u at any point
of the line will be proportional to the distance between the point
(x, y, z) and the point (x0, y0, s0) at which the generator meets the
curve F. It is evident from the formulae (25) that D = 0, that Z>'
is independent of u, and that D" is a polynomial of the second
degree in u:
D" —
+
Since dv is a factor of (24), one system of asymptotic lines consists
of the rectilinear generators v — const. Dividing by dv, the remain
ing differential equation for the other system of asymptotic lines is
of the form
CL tl>
(26) - + Lu2 + Mu + N = 0,
dv
where L, M, and N are functions of the single variable v. An equa
tion of this type possesses certain remarkable properties, which we
shall study later. For example, we shall see that the anharmonic
ratio of any four solutions is a constant. It follows that the anhar
monic ratio of the four points in which a generator meets any four
asymptotic lines of the other system is the same for all generators,
which enables us to discover all the asymptotic lines of the second
system whenever any three of them are known. We shall also
see that whenever one or two integrals of the equation (26) are
known, all the rest can be found by two quadratures or by a single
quadrature. Thus, if all the generators meet a fixed straight line,
that line will be an asymptotic line of the second system, and all
the others can be found by two quadratures. If the surface pos
sesses two such rectilinear directrices, we should know two asymp
totic lines of the second system, and it would appear that another
quadrature would be required to find all the others. But we can
obtain a more complete result. For if a surface possesses two
rectilinear directrices, a protective transformation can be found
which will carry one of them to infinity and transform the surface
into a conoid ; but we saw in § 243 that the asymptotic lines on a
conoid could be found without a single quadrature.
245. Conjugate lines. Any two conjugate diameters of the indica-
trix at a point of a given surface S are called conjugate tangents.
To every tangent to the surface there corresponds a conjugate
tangent, which coincides with the first when and only when the given
512 SURFACES [XII, §245
tangent is an asymptotic tangent. Let z = F(x, y) be the equation of
the surface S, and let m and m' be the slopes of the projections of
two conjugate tangents on the xy plane. These projections on the
xy plane must be harmonic conjugates with respect to the projec
tions of the two asymptotic tangents at the same point of the sur
face. But the slopes of the projections of the asymptotic tangents
satisfy the equation
r + 2s p. + tp.2 — 0 .
In order that the projections of the conjugate tangents should be
harmonic conjugates with respect to the projections of the asymp
totic tangents, it is necessary and sufficient that we should have
(27) r + s (m + m') + tmm' = 0 .
If C be a curve on the surface S, the envelope of the tangent
plane to S at points along this curve is a developable surface which
is tangent to S all along C. At every point M of C the generator of
this developable is the conjugate tangent to the tangent to C. Along
C, x, y, 2, p, and q are functions of a single independent variable a.
The generator of the developable is defined by the two equations
Z - z - p(X - x) - q(Y - y) = 0,
— dz + p dx + q dy — dp (X — x~) — dq( Y — y} = 0 ,
the last of which reduces to
Y — y _ dp rdx + sdy
X — x dq s dx + tdy
Let m be the slope of the projection of the tangent to C and m' the
slope of the projection of the generator. Then we shall have
*y = m y-y = m>
dx X — x
and the preceding equation reduces to the form (27), which proves
the theorem stated above.
Two one-parameter families of curves on a surface are said to
form a conjugate network if the tangents to the two curves of the
two families which pass through any point are conjugate tangents
at that point. It is evident that there are an infinite number of
conjugate networks on any surface, for the first family may be
assigned arbitrarily, the second family then being determined by a
differential equation of the first order.
XII, §245] ASYMPTOTIC LINES CONJUGATE LINES 513
Given a surface represented by equations of the form (19), let us find the
conditions under which the curves u = const, and v = const, form a conjugate
network. If we move along the curve v = const. , the characteristic of the
tangent plane is represented by the two equations
A(X - x) + B(Y -y) + C(Z -z) = 0,
In order that this straight line should coincide with the tangent to the curve
u = const., whose direction cosines are proportional to dx/dv, dy/dv, dz/dv, it
is necessary and sufficient that we should have
cv dv dv
dA dx dB dy dC dz _
du dv du dv du dv
Differentiating the first of these equations with regard to u, we see that the
second may be replaced by the equation
(28)
dudv
dudv
and finally the elimination of A, JB, and C between the equations (21) and (28)
leads to the necessary and sufficient condition
dx dy dz
du du du
dx dy dz
dv dv dv
du dv cu dv du dv
This condition is equivalent to saying that x, y, z are three solutions of a
differential equation of the form
= 0.
(29)
du dv
du
where M and N are arbitrary functions of u and v. It follows that the knowl
edge of three distinct integrals of an equation of this form is sufficient to
determine the equations of a surface which is referred to a conjugate network.
For example, if we set M = N = 0, every integral of the equation (29) is
the sum of a function of u and a function of v ; hence, on any surface whose
equations are of the form
(30) x=/(u)+/i(t>), y = *(u)+^(w), « = ^(u) + ^i(t),
the curves (u) and (v) form a conjugate network.
Surfaces of the type (30) are called surfaces of translation. Any such surface
may be described in two different ways by giving one rigid curve F a motion of
translation such that one of its points moves along another rigid curve T'. For,
514 SURFACES [XII, §246
let MQ, MI , M2 , M be four points of the surface which correspond, respectively,
to the four sets of values (u0, »o), (M, v0), (w0, »), (u, v) of the parameters u and u.
By (30) these four points are the vertices of a plane parallelogram. If «0 is fixed
and u allowed to vary, the point MI will describe a curve T on the surface ; like
wise, if M0 is kept fixed and v is allowed to vary, the point Mz will describe
another curve T' on the surface. It follows that we may generate the surface by
giving F a motion of translation which causes the point M% to describe r", or by
giving T' a motion of translation which causes the point MI to describe T. It is
evident from this method of generation that the two families of curves (M) and (v)
are conjugate. For example, the tangents to the different positions of T' at the
various points of T form a cylinder tangent to the surface along F ; hence the
tangents to the two curves at any point are conjugate tangents.
III. LINES OF CURVATURE
246. Definition and properties of lines of curvature. A curve on a
given surface S is called a line of curvature if the normals to the
surface along that curve form a developable surface. If z — f(x, y)
is the equation of the surface referred to a system of rectangular
axes, the equations of the normal to the surface are
Y=-qZ +(y+qz).
The necessary and sufficient condition that this line should describe
a developable surface is that the two equations
- Z dq + d(y + qz) = 0
should have a solution in terms of Z (§ 223), that is, that we
should have
d(x + pz} _ d(y + qz)
dp dq
or, more simply,
dx + p dz dy -\- q dz
dp dq
Again, replacing dz, dp, and dq by their values, this equation may
be written in the form
(l + p*)dx + pqdy _ pqdx
rdx + sdy sdx + tdy
This equation possesses two solutions in dy/dx which are always
real and unequal if the surface is real, except at an umbilic. For,
if we replace dx and dy by a and /3, respectively, the preceding
XII, §246] LINES OF CURVATURE 515
equation coincides with the equation found above [(14), § 241] for
the determination of the lines of intersection of the principal normal
sections with the tangent plane. It follows that the tangents to the
lines of curvature through any point coincide with the axes of the
indicatrix. We shall see in the study of differential equations that
there is one and only one line of curvature through every non-
singular point of a surface tangent to each one of the axes of the
indicatrix at that point, except at an umbilic. These lines are
always real if the surface is real, and the network which they form
is at once orthogonal and conjugate, — a characteristic property.
Example. Let us determine the lines of curvature of the paraboloid z =
xy/a. In this example
a a a
and the differential equation (33) is
(a2 + 2/2)dx2 = (a2 + z2)dy2 or dx ± dy =Q
Vz2 + a2 Vy2 + a2
If we take the positive sign for both radicals, the general solution is
(x + Vz2 + a'2)(y 4- Vy2 + a2 ) = C,
which gives one system of lines of curvature. If we set
(34) X - z Vy2 + a2 + y Vx2 + a2,
the equation of this system may be written in the form
X + VX2 + a* = C
by virtue of the identity
(z Vy2 + a2 + y Vz2 + a2)2 + a* = [xy + V(x2 + a2)(j/2 + a2)]2
It follows that the projections of the lines of curvature of this first system are
represented by the equation (34), where X is an arbitrary constant. It may be
shown in the same manner that the projections of the lines of curvature of the
other system are represented by the equation
(35) z Vy2 + a2 - y Vx2 + a2 = /*.
From the equation xy = az of the given paraboloid, the equations (34) and
(35) may be written in the form
Vx2 + z2 + Vy2 -f z2 = C, Vx2 + z2 - Vy2 + z2 = C' .
But the expressions Vz2 + z2 and Vy2 + z2 represent, respectively, the dis
tances of the point (z, y, z) from the axes of z and y. It follows that the lines
of curvature on the paraboloid are those curves for which the sum or the difference
of the distances of any point upon them from the axes of x and y is a constant.
516 SURFACES [XII, §247
247. Evolute of a surface. Let C be a line of curvature on a sur
face S. As a point M describes the curve C, the normal MN to the
surface remains tangent to a curve T. Let (X, Y, Z) be the coor
dinates of the point A at which MN is tangent to T. The ordinate
Z is given by either of the equations (32), which reduce to a single
e< {nation since C is a line of curvature. The equations (32) may
be written in the form
z _z _ (1 + P2) dx + pq dy ^pgdx + (l+ ?2) dy
r dx + s dy s dx + tdy
Multiplying each term of the first fraction by dx, each term of the
second by dy, and then taking the proportion by composition, we
find
_ dx2 + dy* + (p dx + g dy}*
r dx2 + 2s dx dy + t dif
Again, since dx, dtj, and dz are proportional to the direction cosines
a, ft, y of the tangent, this equation may be written in the form
_
£* *5 -
_ a2 + /?2 + (pa +
-
m2 + 2s«p + tfi2 ra2 + 2safi
Comparing this formula with (7), which gives the radius of curva
ture R of the normal section tangent to the line of curvature, with
the proper sign, we see that it is equivalent to the equation
(36) Z - z = R = RV,
where v is the cosine of the acute angle between the z axis and the
positive direction of the normal. But z + Kv is exactly the value
of Z for the center of curvature of the normal section under con
sideration. It follows that the point of tangency A of the normal
MN to its envelope T coincides with the center of curvature of the
principal normal section tangent to C at M. Hence the curve F is
the locus of these centers of curvature. If we consider all the lines
of curvature of the system to which C belongs, the locus of the cor
responding curves r is a surface 2 to which every normal to the
given surface S is tangent. For the normal MN, for example, is
tangent at A to the curve r which lies on 2.
The other line of curvature C' through M cuts C at right angles.
The normal to S along C' is itself always tangent to a curve T'
which is the locus of the centers of curvature of the normal sections
XII, § 248]
LINES OF CURVATURE
517
tangent to C". The locus of this curve T' for all the lines of curva
ture of the system to which C' belongs is a surface 2' to which all
the normals to S are tangent. The two surfaces 2 and 2' are not
usually analytically distinct, but form two nappes of the same sur
face, which is then represented by an irreducible equation.
The normal MN to S is tangent to each of these nappes 2 and 2'
at the two principal centers of curvature A and A ' of the surface S
at the point M. It is easy to find the tangent
planes to the two nappes at the points A and A'
(Fig. 51). As the point M describes the curve
C, the normal MN describes the developable
surface D whose edge of regression is F ; at
the same time the point A' where MN touches
2' describes a curve y' distinct from Ff, since
the straight line MN cannot remain tangent to
two distinct curves F and F'. The developable
D and the surface 2' are tangent at A ' ; hence
the tangent plane to 2' at A' is tangent to D
all along MN. It follows that it is the plane
NMT, which passes through the tangent to C.
Similarly, it is evident that the tangent plane
to 2 at A is the plane NMT' through the tan
gent to the other line of curvature C'.
The two planes NMT and NMT' stand at right angles. This fact
leads to the following important conception. Let a normal OM be
dropped from any point O in space on the surface S, and let A and
A' be the principal centers of curvature of S on this normal. The
tangent planes to 2 and 2' at A and A ', respectively, are perpendic
ular. Since each of these planes passes through the given point 0, it
is clear that the two nappes of the e volute of any surface S, observed
from any point O in space, appear to cut each other at right angles.
The converse of this proposition will be proved later.
248. Rodrigues' formulae. If X, /A, v denote the direction cosines
of the normal, and R one of the principal radii of curvature, the
corresponding principal center of curvature will be given by the
formulae
FIG. 51
(37) X =
Z = z + Rv.
As the point (x, y, z) describes a line of curvature tangent to
the normal section whose radius of curvature is R, this center of
518 SURFACES [XII, §249
curvature, as we have just seen, will describe a curve F tangent to
the normal MN; hence we must have
dX _dY _dZ
X fji v
or, replacing X, Y, and Z by their values from (37) and omitting the
common term dR,
dx -f R d\ _ dy + Rdp. _ dz + Rdy
X p. v
The value of any of these ratios is zero, for if we take them by
composition after multiplying each term of the first ratio by X, of
the second by p., and of the third by v, we obtain another ratio
equal to any of the three ; but the denominator of the new ratio is
unity, while the numerator
X dx + fj. dy +• v dz + R(\ dX + p. dp. + v dv)
is identically zero. This gives immediately the formulae of Olinde
Rodrigues :
(38) dx + R dX = 0, dy + Rdn = 0, dz + Rdv = 0,
which are very important in the theory of surfaces. It should be
noticed, however, that these formulae apply only to a displacement
of the point (a;, y, z) along a line of curvature.
249. Lines of curvature in parameter form. If the equations of the
surface are given in terms of two parameters u and v in the form
(19), the equations of the normal are
X—x _ Y—y_Z—z
A B C
where A, B, and C are determined by the equations (21). The
necessary and sufficient condition that this line should describe a
developable surface is, by § 223,
(39)
dx dy dz
ABC
dA dB dC
= 0,
where x, y, z, A, B, and C are to be replaced by their expressions
in terms of the parameters u and v; hence this is the differential
equation of the lines of curvature.
XII, §249] LINES OF CURVATURE 519
As an example let us find the lines of curvature on the helicoid
z = a arc tan - ,
x
whose equation is equivalent to the system
In this example the equations for A, B, and C are
A cos 0 + B sin 0 = 0 , — Ap sin 0 + Bp cos 0 -f Ca = 0 .
Taking C = p, we find A = a sin 0, B = — a cos 0. After expansion and simpli
fication the differential equation (39) becomes
0 or de-± dp
Choosing the sign + , for example, and integrating, we find
p + V/o2 + a2 = ae»-»o, or p = - [e«-ec - e~(«- »o)] .
2
The projections of these lines of curvature on the xy plane are all spirals which
are easily constructed.
The same method enables us to form the equation of the second
degree for the principal radii of curvature. With the same symbols
A, B, C, \, n, v we shall have, except for sign,
f B2 + C2
We shall adopt as the positive direction of the normal that which
is given by the preceding equations. If .R is a principal radius of
curvature, taken with its proper sign, the coordinates of the corre
sponding center of curvature are
where
R = p^A* + B2 + C2.
If the point (x, y, z) describes the line of curvature tangent to the
principal normal section whose radius of curvature is R, we have
seen that the point (X, Y, Z) describes a curve F which is tangent
to the normal to the surface. Hence we must have
dx + p dA + A dp _ dy + p dB + B dp _ dz + pdC + Cdp
A B C '
620
SURFACES
[XII, § 250
or, denoting the common values of these ratios by dp + K,
(dx + pdA — A K — 0 ,
dy + pdB - BK= 0,
dz + pdC - CK= 0.
Eliminating p and K from these three equations, we find again the
differential equation (39) of the lines of curvature. But if we
replace dx, dy, dz, dA, dB, and dC by the expressions
ox
^— d u
du
dx
^—
dv
dv,
dC T dC
— du Hh a~
tin cv
respectively, and then eliminate du, dv, and K, we find an equation
for the determination of p :
(41)
dx
dA
dx
dA
du
+ P~du
fa
+ p^
A
du
+ p°du
dv
cv
B
o ^
dc
dz
dC
—
—
C
du
du
du
cv
= 0.
If we replace p by R/ ~vA* + B'z -f C2, this equation becomes an
equation for the principal radii of curvature.
The equations (39) and (41) enable us to answer many questions
which we have already considered. For example, the necessary
and sufficient condition that a point of a surface should be a para
bolic point is that the coefficient of p2 in (41) should vanish. In
order that a point be an umbilic, the equation (39) must be satisfied
for all values of du and dv
As an example let us find the principal radii of curvature of the rectilinear
helicoid. With a slight modification of the notation used above, we shall have
in this example
« = MCOS«, y— u sin v, z = av,
A = as'mv, B = —acosv, C = u,
and the equation (41) becomes
whence R — ± (a2 + u2)/a. Hence the principal radii of curvature of the helicoid
are numerically equal and opposite in sign.
250. Joachimsthal's theorem. The lines of curvature on certain
surfaces may be found by geometrical considerations. For example,
it is quite evident that the lines of curvature on a surface of revolu
tion are the meridians and the parallels of the surface, for each of
XII, §251] LINES OF CURVATURE 521
these curves is tangent at every point to one of the axes of the
indicatrix at that point. This is again confirmed by the remark
that the normals along a meridian form a plane, and the normals
along a parallel form a circular cone, — in each case the normals
form a developable surface.
On a developable surface the first system of lines of curvature
consists of the generators. The second system consists of the
orthogonal trajectories of the generators, that is, of the involutes of
the edge of regression (§ 231). These can be found by a single quad
rature. If we know one of them, all the rest can be found without
even one quadrature. All of these results are easily verified directly.
The study of the theory of evolutes of a skew curve led Joa-
chimsthal to a very important theorem, which is often used in that
theory. Let S and S' be two surfaces whose line of intersection C
is a line of curvature on each surface. The normal MN to S along
C describes a developable surface, and the normal MN' to S' along
C describes another developable surface. But each of these normals
is normal to C. It follows from § 231 that if two surfaces have a
common line of curvature, they intersect at a constant angle along
that line.
Conversely, if two surfaces intersect at a constant angle, and if
their line of intersection is a line of curvature on one of them, it is
also a line of curvature on the other. For we have seen that if one
family of normals to a skew curve C form a developable surface,
the family of normals obtained by turning each of the first family
through the same angle in its normal plane also form a developable
surface.
Any curve whatever on a plane or on a sphere is a li»e of curva
ture on that surface. It follows as a corollary to Joachimsthal's
theorem that the necessary and sufficient condition that a plane curve
or a spherical curve on any surface should be a line of curvature is
that the plane or the sphere on which the curve lies should cut the
surface at a constant angle.
251. Dupin's theorem. We have already considered [§§ 43, 146]
triply orthogonal systems of surfaces. The origin of the theory of
such systems lay in a noted theorem due to Dupin, which we shall
proceed to prove :
Given any three families of surfaces which form a triply orthogonal
system : the intersection of any two surfaces of different families is a
line of curvature on each of them.
522 SURFACES [XII, §251
We shall base the proof on the following remark. Let F(x, y, z) = 0
be the equation of a surface tangent to the xij plane at the origin. Then
we shall have, for x — y = z — 0, dF/dx = 0, dF/cy = 0, but cF/dz does
not vanish, in general, except when the origin is a singular point.
It follows that the necessary and sufficient condition that the x and
y axes should be the axes of the indicatrix is that s = 0. But the
value of this second derivative s = c2z/cx dy is given by the equation
r\ f\ I r\ o Y I O Ci J"^ I O O V 7 ' O
£X <7y CX CZ Cy CZ CZ* CZ
Since p and q both vanish at the origin, the necessary and sufficient
condition that s should vanish there is that we should have
dx dy
Now let the three families of the triply orthogonal system be given
by the equations
where Fl , F2, F3 satisfy the relation
f A <3\ 1 ? J _1 2 i 1 2 — f\
(4:0) -^ o I a 1 < ^ " — ^
cx ox cy cy cz cz
and two other similar relations obtained by cyclic permutation of
the subscripts 1, 2, 3. Through any point M in space there passes,
in general, one surface of each of the three families. The tangents to
the three curves of intersection of these three surfaces form a trirec-
tangular trihedron. In order to prove Dupin's theorem, it will be
sufficient to show that each of these tangents coincides with one of
the axes of the indicatrix on each of the surfaces to which it is
tangent.
In order to show this, let us take the point M as origin and the
edges of the trirectangular trihedron as the axes of coordinates ;
then the three surfaces pass through the origin tangent, respec
tively, to the three coordinate planes. At the origin we shall have,
for example,
l&Y-o.
§Uo, I^Uo, l^) = o.
XII, §251] LINES OF CURVATURE 523
The axes of x and y will be the axes of the indicatrix of the surface
F(x, y, z) = 0 at the origin if (c^F^/dx dy)0 = 0. To show that this
is the case, let us differentiate (43) with respect to y, omitting the
terms which vanish at the origin ; we find
££\ / 1^\ /d_F\\
e \dx di/ \ dz /0 \dy
or
From the two relations analogous to (43) we could deduce two
equations analogous to (44), which may be written down by cyclic
permutation :
o _
;
From (44) and (45) it is evident that we shall have also
which proves the theorem.
A remarkable example of a triply orthogonal system is furnished
by the confocal quadrics discussed in § 147. It was doubtless the
investigation of this particular system which led Dupin to the gen
eral theorem. It follows that the lines of curvature on an ellipsoid
or an hyperboloid (which had been determined previously by Monge)
are the lines of intersection of that surface with its confocal quadrics.
The paraboloids represented by the equation
.1 o
y , * - o~ _
,
p — X (/ — A
where X is a variable parameter, form another triply orthogonal
system, which determines the lines of curvature on the paraboloid.
Finally, the system discussed in § 24G,
= y,
is triply orthogonal.
524 SURFACES [XII, §252
The study of triply orthogonal systems is one of the most interest
ing and one of the most difficult problems of differential geometry.
A very large number of memoirs have been published on the subject,
the results of which have been collected by Darboux in a recent
work.* Any surface S belongs to an infinite number of triply
orthogonal systems. One of these consists of the family of surfaces
parallel to S and the two families of developables formed by the
normals along the lines of curvature on S. For, let O be any point
on the normal MN to the surface S at the point M, and let MT
and MT' be the tangents to the two lines of curvature C and C"
which pass through M; then the tangent plane to the parallel sur
face through O is parallel to the tangent plane to S at M, and the
tangent planes to the two developables described by the normals to
S along C and C' are the planes MNT and MNT', respectively. These
three planes are perpendicular by pairs, which shows that the system
is triply orthogonal.
An infinite number of triply orthogonal systems can be derived
from any one known triply orthogonal system by means of succes
sive inversions, since any inversion leaves all angles unchanged.
Since any surface whatever is a member of some triply orthogonal
system, as we have just seen, it follows that an inversion carries the
lines of curvature on any surface over into the lines of curvature on
the transformed surface. It is easy to verify this fact directly.
252. Applications to certain classes of surfaces. A large number of problems
have been discussed in which it is required to find all the surfaces whose lines
of curvature have a preassigned geometrical property. AVe shall proceed to
indicate some of the simpler results.
First let us determine all those surfaces for which one system of lines of
curvature are circles. By Joachimsthal's theorem, the plane of each of the
circles must cut the surface at a constant angle. Hence all the normals to the
surface along any circle C of the system must meet the axis of the circle, i.e.
the perpendicular to its plane at its center, at the same point 0. The sphere
through C about 0 as center is tangent to the surface all along C ; hence the
required surface must be the envelope of a one-parameter family of spheres.
Conversely, any surface which is the envelope of a one-parameter family of
spheres is a solution of the problem, for the characteristic curves, which are
circles, evidently form one system of lines of curvature.
Surfaces of revolution evidently belong to the preceding class. Another
interesting particular case is the so-called tubular surface, which is the envelope
of a sphere of constant radius whose center describes an arbitrary curve F. The
characteristic curves are the circles of radius R whose centers lie on r and
whose planes are normal to T. The normals to the surface are also normal to T ;
* Lemons sur les systemes orthogonaux ft les coordonntes curvilignes, 1898.
XII, §252] LINES OF CURVATURE 525
hence the second system of lines of curvature are the lines in which the surface
is cut by the developable surfaces which may be formed from the normals to r.
If both systems of lines of curvature on a surface are circles, it is clear from
the preceding argument that the surface may be thought of as the envelope of
either of two one-parameter families of spheres. Let <Si , S2 , S3 be any three
spheres of the first family, C\ , C2 , C3 the corresponding characteristic curves,
and MI , M2 , Ms the three points in which Ci , C2 , C3 are cut by a line of curva
ture C" of the other system. The sphere <S' which is tangent to the surface along
C" is also tangent to the spheres Si , S2 , Ss at MI , JV/2 , M3 , respectively. Hence
the required surface is the envelope of a family of spheres each of which touches
three fixed spheres. This surface is the well-known Dupin cyclide. Mannheim
gave an elegant proof that any Dupin cyclide is the surface into which a certain
anchor ring is transformed by a certain inversion. Let 7 be the circle which
is orthogonal to each of the three fixed spheres Si, S2, 83. An inversion whose
pole is a point on the circumference of 7 carries that circle into a straight line
00', and carries the three spheres Si, S2, Ss into three spheres 2i, 22, 23
orthogonal to OO7, that is, the centers of the transformed spheres lie on OO'.
Let Ci, C2, C£ be the intersections of these spheres with any plane through
O(y, C' a circle tangent to each of the circles C{, C2, €3, and 2' the sphere
on which C' is a great circle. It is clear that 2' remains tangent to each of the
spheres Si, S2, 23 as the whole figure is revolved about 00', and that the
envelope of 2' is an anchor ring whose meridian is the circle C'.
Let us now determine the surface for which all of the lines of curvature of
one system are plane curves whose planes are all parallel. Let us take the xy
plane parallel to the planes in which these lines of curvature lie, and let
x cos a + y sin a = F(a, z)
be the tangential equation of the section of the surface by a parallel to the xy
plane, where F(a, z) is a function of a and z which depends upon the surface
under consideration. The coordinates x and y of a point of the surface are
given by the preceding equation together with the equation
dF
— x sin a + y cos a =
da
The formulae for x, y, z are
dF d F
(46) z = Fcos<r sin a, y = Fs'ma-\ cos a, z = z.
da da
Any surface may be represented by equations of this form by choosing the
function F(a, z) properly. The only exceptions are the ruled surfaces whose
directing plane is the xy plane. It is easy to show that the coefficients A, B, C
of the tangent plane may be taken to be
dF
A = cos a , .B = sin a . C = ;
dz
hence the cosine of the angle between the normal and the z axis is
In order that all the sections by planes parallel to the xy plane be lines of curva
ture, it is necessary and sufficient, by Joachimsthal's theorem, that each of
526 SURFACES [XII,§253
these planes cut the surface at a constant angle, i.e. that v be independent of a.
This is equivalent to saying that Fz(a, z) is independent of a, i.e. that F(a, z)
is of the form
F(a, z) = t(z) + f (a) ,
where the functions 0 and \f/ are arbitrary. Substituting this value in (46), we
Bee that the most general solution of the problem is given by the equations
( x = f(a) cos a — ^'(«) sin a + <f>(z) cos a ,
(47) -| y = \j/(a) sin a + f (a) cos a + <£(z) sin a ,
These surfaces may be generated as follows. The first two of equations (47),
for z constant and a variable, represent a family of parallel curves which are
the projections on the xy plane of the sections of the surface by planes parallel
to the xy plane. But these curves are all parallel to the curve obtained by set
ting <f>(z) = 0. Hence the surfaces may be generated as follows : Taking in the
xy plane any curve whatever and its parallel curves, lift each of the curves verti
cally a distance given by some arbitrary law ; the curves in their new positions form
a surface which is the most general solution of the problem.
It is easy to see that the preceding construction may be replaced by the
following : The required surfaces are those described by any plane curve whose
plane rolls without slipping on a cylinder of any base. By analogy with plane
curves, these surfaces may be called rolled surfaces or roulettes. This fact may
be verified by examining the plane curves a = const. The two families of lines
of curvature are the plane curves z = const, and a = const.
IV. FAMILIES OF STRAIGHT LINES
The equations of a straight line in space contain four variable
parameters. Hence we may consider one-, two-, or three-parameter
families of straight lines, according to the number of given relations
between the four parameters. A one-parameter family of straight
lines form a ruled surface. A two-parameter family of straight
lines is called a line congruence, and, finally, a three-parameter
family of straight lines is called a line complex.
253. Ruled surfaces. Let the equations of a one-parameter family
of straight lines (G) be given in the form
(48) x = az+p, y = bz + q,
where a, b, p, q are functions of a single variable parameter u. Let
us consider the variation in the position of the tangent plane to the
surface S formed by these lines as the point of tangency moves along
any one of the generators G. The equations (48), together with the
equation z = z, give the coordinates x, y, z of a point M on S in terms
XII, §253] FAMILIES OF STRAIGHT LINES 527
of the two parameters z and u ; hence, by § 39, the equation of the
tangent plane at M is
X — x Y — y Z — z
a b 1=0,
a'z+p' b'z + q' 0
where a', b', p', q' denote the derivatives of a, b, p, q with respect
to u. Eeplacing x and y by az -f p and bz + q, respectively, and
simplifying, this equation becomes
(49) (b'z + q'}(X - aZ -p) - (a'z + p')(Y - bZ - q) = 0.
In the first place, we see that this plane always passes through the
generator G, which was evident a priori, and moreover, that Jthe plane
turns around G as the point of tangency M moves along G, at least
unless the ratio (a'z + p')/(b'z + <?') is independent of z, i.e. unless
a'q' — b'p' = 0, — we shall discard this special case in what follows.
Since the preceding ratio is linear in z, every plane through a gen
erator is tangent to the surface at one and only one point. As the
point of tangency recedes indefinitely along the generator in either
direction the tangent plane P approaches a limiting position P',
which we shall call the tangent plane at the point at infinity on that
generator. The equation of this limiting plane P' is
(50) b'(X- aZ -p)-a'(Y-bZ -q) = 0.
Let w be the angle between this plane P' and the tangent plane P at
a point M (x, y, z) of the generator. The direction cosines (a1, ft', y')
and (a, /?, y) of the normals to P' and P are proportional to
b', - a', a'b - ab'
and
b'z + q', -(a'z+p1), b(a'z+p')-a(b'z + q'),
respectively; hence
Az + B
cos u> = aa' + (3/3' + yy = —j=^
Az* + 2Bz + C
where
A =a'2 +b'2 +(ab' -ba'Y,
B = a'p' + b'q' + (ab' - ba'}(aq' - bp') ,
C=p'2 +q12 +(aq'-bp'y.
After art easy reduction, we find, by Lagrange's identity (§ 131),
- B2 (a'q' - b'p'} Vl -f a* +
(51) tan CD = — - — = •» — —
' • Az + B Az + B
528 SURFACES [xii, § 253
It follows that the limiting plane P' is perpendicular to the tangent
plane Pv at a point Ol of the generator whose ordinate z{ is given by
the formula
_B_ a'p' + b'g' + (aV - ba')(ag' - bp'~)
KI~ A~ a'2 + b'2 + (ab' -baj2
The point 0± is called the central point of the generator, and the tan
gent plane P! at Ot is called the central plane. The angle 6 between
the tangent plane P at any point M of the generator and this central
plane Pl is Tr/2 — o>, and the formula (51) may be replaced by the
formula
tan o »
b'p1) Vl
Let p be the distance between the central point O^ and the point M,
taken with the sign + or the sign — according as the angle which
Oi M makes with the positive z axis is acute or obtuse. Then we
shall have p = (z — «t) Vl -f a2 -j- b'2, and the preceding formula may
be written in the form
(53) tan0 = fy,
where k, which is called the parameter of distribution, is defined by
the equation
= a>* >* <-
The formula (53) expresses in very simple form the manner in which
the tangent plane turns about the generator. It contains no quantity
which does not have a geometrical meaning : we shall see presently
that k may be defined geometrically. However, there remains a cer
tain ambiguity in the formula (53), for it is not immediately evident
in which sense the angle 0 should be counted. In other words, it is
not clear, a priori, in which direction the tangent plane turns around
the generator as the point moves along the generator. The sense of
this rotation may be determined by the sign of k.
In order to see the matter clearly, imagine an observer lying on a
generator G. As the point of tangency M moves from his feet toward
his head he will see the tangent plane P turn either from his left
to his right or vice versa. A little reflection will show that the
sense of rotation defined in this way remains unchanged if the
observer turns around so that his head and feet change places.
Two hyperbolic paraboloids having a generator in common and
XII, §253] FAMILIES OF STRAIGHT LINES 529
lying symmetrically with respect to a plane through that generator
give a clear idea of the two possible situations. Let us now move
the axes in such a way that the new origin is at the central point Ol,
the new z axis is the generator G itself, and the xz plane is the cen
tral plane Pt. It is evident that the value of the parameter of dis
tribution (54) remains unchanged during this movement of the axes,
and that the formula (53) takes the form
(53') tan0 = &*,
where 6 denotes the angle between the xz plane Pl and the tangent
plane P, counted in a convenient sense. For the value of u0 which
corresponds to the z axis we must have a = b = p — q = 0, and the
equation of the tangent plane at any point M of that axis becomes
(b'e + q')X-(a'z+p')Y=Q.
In order that the origin be the central point and the xz plane the
central plane, we must have also a' = 0, q' = 0 ; hence the equation
of the tangent plane reduces to Y = (b'z/p'^X, and the formula (54)
gives k=—b'/p'. It follows that the angle 0 in (53') should be
counted positive in the sense from Oy toward Ox. If the orienta
tion of the axes is that adopted in § 228, an observer lying in the
z axis will see the tangent plane turn from his left toward his right
if k is positive, or from his right toward his left if k is negative.
The locus of the central points of the generators of a ruled surface
is called the line of strict ion. The equations of this curve in terms
of the parameter u are precisely the equations (48) and (52).
Note. If a'q' = b'p' for a generator G, the tangent plane is the
same at any point of that generator. If this relation is satisfied
for every generator, i.e. for all values of u, the ruled surface is a
developable surface (§ 223), and the results previously obtained can
be easily verified. For if a' and b' do not vanish simultaneously,
the tangent plane is the same at all points of any generator G,
and becomes indeterminate for the point z = — p'/a' =±= — q'/b', i.e.
for the point where the generator touches its envelope. It is easy
to show that this value for z is the same as that given by (52) when
a'q' = b'p'. It follows that the line of striction becomes the edge
of regression on a developable surface. The parameter of distribution
is infinite for a developable.
If a' = b' = 0 for every generator, the surface is a cylinder and
the central point is indeterminate.
530 SURFACES [XII, §254
254. Direct definition of the parameter of distribution. The central
point and the parameter of distribution may be defined in an entirely
different manner. Let G and G^ be two neighboring generators cor
responding to the values u and u -f- h of the parameter, respectively,
and let GI be given by the equations
(55) x — (a + Aa) z + p -f Aj5, y = (b + Aft) z + q + Ay.
Let 8 be the shortest distance between the two lines G and Gl , a the
angle between G and Gl} and (X, Y, Z) the point where G meets the
common perpendicular. Then, by well-known formulae of Analytic
Geometry, we shall have .
_ _ Aa Ay + Aft Aj? + (a Aft — 6 Aa)[(a + Aa) Ay — (6 + Aft)Ap]
(Aa)2 + (Aft)2 + (a Aft - 6 Aa)2
- _ Aa Ay — Aft Ap
V(Aa)2 + (Aft)2 + (a Aft - & Aa)2'
V(Aa)2 + (Aft)2 + (« Aft - 6 Aa)2
sin a = — . v = — ,
Va2 + b2 + 1 V(a -f- Aa)2 + (b + Aft)2 + 1
As h approaches zero, Z approaches the quantity «x defined by (52),
and (sin <*)/8 approaches k. Hence the central point is the limiting
position of the foot of the common perpendicular to G and Gl, while
the parameter of distribution is the limit of the ratio (sin a)/8.
In the expression for 8 let us replace Aa, A&, A/?, Ay by their
expansions in powers of h:
h2
Aa = ha' + — a" -\ ----
and the similar expansions for Aft, A/?, Ay. Then the numerator of
the expression for 8 becomes
while the denominator is always of the first order with respect
to h. It is evident that 8 is in general an infinitesimal of the first
order with respect to h, except for developable surfaces, for which
a'y'= b'j)'- But the coefficient of hs/2 is the derivative of a'q' — b'p';
hence this coefficient also vanishes for a developable, and the shortest
distance between two neighboring generators is of the third order
(§ 230). This remark is due to Bouquet, who also showed that if
this distance is constantly of the fourth order, it must be precisely
zero; that is, that in that case the given straight lines are the
XII, §255] FAMILIES OF STRAIGHT LINES 531
tangents to a plane curve or to a conical surface. In order to prove
this, it is sufficient to carry the development of Aa Ay — Aft A/> to
terms of the fourth order.
255. Congruences. Focal surface of a congruence. Every two-parameter
family of straight lines
(56) x = az+p, y = bz + q,
where a, b, p, q depend on two parameters a and ft, is called a line
congruence. Through any point in space there pass, in general, a
certain number of lines of the congruence, for the two equations (56)
determine a certain number of definite sets of values of a and ft when
a-, y, and z are given definite values. If any relation between a and ft
be assumed, the equations (56) will represent a ruled surface, which
is not usually developable. In order that the surface be developable,
we must have
da dq — db dp = 0,
or, replacing da by (da/da) da -f (da/d/3) dft, etc.,
d n r* 112 ' ^ n
(*tr\ \ ^~ ^ ' ^ ^
(57) '** n
' dft ^}\ca ' dft
This is a quadratic equation in dft/da. Solving it, we should usu
ally obtain two distinct solutions,
dft dft
(oo) — — = \j/i (cr, ft) , — = \I/2 (<*) p) ,
«<x rtnr
either of which defines a developable surface. Under very gen
eral limitations, which we shall state precisely a little later and
which we shall just now suppose fulfilled, each of these equations
is satisfied by an infinite number of functions of a, and each of them
has one and only one solution which assumes a given value ft0 when
a = a0. It follows that every straight line G of the congruence
belongs to two developable surfaces, all of whose generators are
members of the congruence. Let F and F' be the edges of regression
of these two developables, and A and A' the points where G touches
F and F', respectively. The two points A and A' are called the focal
points of the generator G. They may be found as follows without
integrating the equation (57). The ordinate z of one of these points
must satisfy both of the equations
z da -f- dp = 0 , .~ db + dq = 0 ,
532 SURFACES [XII, §255
or, replacing da, db, dp, dy by their developments,
Eliminating z between these two equations, we find again the equa
tion (57). But if we eliminate dp /da we obtain an equation of the
second degree
whose two solutions are the values of z for the focal points.
The locus of the focal points A and A' consists of two nappes
2 and 2' of a surface whose equations are given in parameter form
by the formulae (56) and (59). These two nappes are not in general
two distinct surfaces, but constitute two portions of the same ana
lytic surface. The whole surface is called the focal surface. It is
evident that the focal surface is also the locus of the edges of regres
sion of the developable surfaces which can be formed from the lines
of the congruence. For by the very definition of the curve T the
tangent at any point a is a line of the congruence; hence a is a
focal point for that line of the congruence. Every straight line
of the congruence is tangent to each of the nappes 2 and 2', for it
is tangent to each of two curves which lie on these two nappes,
respectively.
By an argument precisely similar to that of § 247 it is easy to
determine the tangent planes at A and A' to 2 and 2' (Fig. 51).
As the line G moves, remaining tangent to r, for example, it also
remains tangent to the surface 2'. Its point of tangency A ' will
describe a curve y' which is necessarily distinct from r'. Hence
the developable described by G during this motion is tangent to 2'
at A ', since the tangent planes to the two surfaces both contain the
line G and the tangent line to y'. It follows that the tangent plane
to 2' at A' is precisely the osculating plane of r at A. Likewise,
the tangent plane to 2 at A is the osculating plane of T at A'.
These two planes are called the focal planes of the generator G.
It may happen that one of the nappes of the focal surface degen
erates into a curve C. In that case the straight lines of the con
gruence are all tangent to 2, and merely meet C. One of the
families of developables consists of the cones circumscribed about 2
XH, § aw] FAMILIES OF STRAIGHT LINES 533
whose vertices are on C. If both of the nappes of the focal surface
degenerate into curves C and C", the two families of developables
consist of the cones through one of the curves whose vertices lie
on the other. If both the curves C arid C' are straight lines, the
congruence is called a linear congruence.
256. Congruence of normals. The normals to any surface evidently
form a congruence, but the converse is not true : there exists no
surface, in general, which is normal to every line of a given con
gruence. For, if we consider the congruence formed by the normals
to a given surface S, the two nappes of the focal surface are evidently
the two nappes 2 and 2' of the e volute of S (§ 247), and we have seen
that the two tangent planes at the points A and A' where the same
normal touches 2 and 2' stand at right angles. This is a character
istic property of a congruence of normals, as we shall see by trying
to find the condition that the straight line (56) should always remain
normal to the surface. The necessary and sufficient condition that it
should is that there exist a function /(a, /3) such that the surface 5
represented by the equations
(60) x = az+p, y = bz + q, z=f(a,p-)
is normal to each of the lines (6'). It follows that we must have
8x . d dz
8x dy dz
adp^bW + W =
or, replacing x and y by az + p and bz + q, respectively, and divid
ing by Va2 + I)'2 + 1,
dp dq
a " t~ " '~
^(gVa« + ft'+l)+ -0;
&* Va2
Va'
=0.
The necessary and sufficient condition that these equations be com
patible is
(62)
534 SURFACES [XII, §256
If this condition is satisfied, z can be found from (61) by a single
quadrature. The surfaces obtained in this way depend upon a con
stant of integration and form a one-parameter family of parallel
surfaces.
In order to find the geometrical meaning of the condition (62), it
should be noticed that that condition, by its very nature, is inde
pendent of the choice of axes and of the choice of the independent
variables. We may therefore choose the z axis as a line of the con
gruence, and the parameters a and ft as the coordinates of the point
where a line of the congruence pierces the xy plane. Then we shall
have p = a, q = ft, and a and b given functions of a and ft which van
ish for a = ft = 0. It follows that the condition of integrability, for
the set of values a = ft = 0, reduces to the equation da/dft = 8b/da.
On the other hand, the equation (57) takes the form
Qj"% **"!•* I I •"- Oy-»llxl-AWAJ '^ !**• — — V/ •
0/3 \tfa <?/?/ da
which is the equation for determining the lines of intersection of
the xy plane with the developables of the congruence after a and
ft have been replaced by « and ?/, respectively. The condition
da/dft = db/da, for a = ft = 0, means that the two curves of this
kind which pass through the origin intersect at right angles ; that
is, the tangent planes to the two developable surfaces of the congru
ence which pass through the z axis stand at right angles. Since the
line taken as the z axis was any line of the congruence, we may state
the following important theorem:
The necessary and sufficient condition that the straight lines of a
given congruence be the normals of some surface is that the focal planes
through every line of the congruence should be perpendicidar to each
other.
Note. If the parameters a and ft be chosen as the cosines of the angles which
the line makes with the x and y axes, respectively, we shall have
a ^
VI + a'2 + 62 =
Vl~- a2 - p Vl - a* - p* Vl - a* -
and the equations (61) become
(63)
Vl- «*-/3V eft dft
XII. § 2,-i7]
FAMILIES OF STRAK1IIT LINES
535
Then the condition of integrability (62) reduces to the form dq/da = dp/ dp, which
means that p and q must be the partial derivatives of the same function F(a, p) :
dF
dF
ejs'
where F(a, p) can be found by a single quadrature. It follows that z is the
solution of the total differential equation
d(-
whence
Badp
dp'2
z = Vl - a2 - p
where C is an arbitrary constant.
C + F- a— -
da
257. Theorem of Malus. If rays of light from a point source are reflected (or
refracted) by any surface, the reflected (or refracted) rays are the normals to
each of a family of parallel surfaces. This theorem, which is due to Malus, has
been extended by Cauchy, Dupin, Gergonne, and Quetelet to the case of any
number of successive reflections or refractions, and we may state the following
more general theorem :
If a family of rays of light are normal to some surface at any time, they retain
that property after any number of reflections and refractions.
Since a reflection may be regarded as a refraction of index — 1, it is evidently
sufficient to prove the theorem for a single refraction. Let S be a surface nor
mal to the unrefracted rays, mM an incident ray which meets the surface of
separation S at a point M, and MR the refracted ray. By Descartes' law, the
incident ray, the refracted ray, and the normal MN lie in a plane, and the
angles i and r (Fig. 52) satisfy the relation
n sin i = sin r. For definiteness we shall sup
pose, as in the figure, that n is less than
unity. Let I denote the distance Mm, and
let us lay off on the refracted ray extended
a length I' = Mm' equal to k times I, where
A; is a constant factor which we shall deter
mine presently. The point m' describes a
surface S'. We shall proceed to show that
k may be chosen in such a way that Mm' is
normal to S'. Let C be any curve on S.
As the point m describes C the point M
describes a curve T on the surface 2, and
the corresponding point m' describes another
curve C" on S'. Let s, cr, s' be the lengths of the arcs of the three curves C, r,
C' measured from corresponding fixed points on those curves, respectively,
w the angle which the tangent M TI to r makes with the tangent MT to the
normal section by the normal plane through the incident ray, and <£ and <p' the
angles which M T\ makes with Mm and Mm', respectively. In order to find
cos0, for example, let us lay off on Mm a unit length and project it upon 3/Tlt
FIG. 52
536 SURFACES [XII, §258
first directly, then by projecting it upon NT and from M T upon M 7\. This,
and the similar projection from Mm' upon -MTi, give the equations
cos <f> = sin i cos w , cos <p' = sin r cos o» .
Applying the formula (10') of § 82 for the differential of a segment to the seg
ments Mm and Mm', we find
dl = — da- cos w sin i ,
dl' = — da- cos u» sin r — ds' cos 0 ,
where 0 denotes the angle between m'M and the tangent to C". Hence, replacing
dl' by k dl, we find
cos di dcr(k sin i — sin r) = ds' cos 0 ,
or, assuming k = n,
ds' cos 0 = 0.
It follows that Mm' is normal to C", and, since C' is any curve whatever on
S', 3fm' is also normal to the surface S'. This surface S' is called the anti-
caustic surface, or the secondary caustic. It is clear that S' is the envelope of
the spheres described about M as center with a radius equal to n times Mm ;
hence we may state the following theorem :
Let us consider the surface S which is normal to the incident rays as the envelope
of a family of spheres whose centers lie on the surface of separation 2. Then the
anticaustic for the refracted rays is the envelope of a family of spheres with the
same centers, whose radii are to the radii of the corresponding spheres of the first
family as unity is to the index of refraction.
This envelope is composed of two nappes which correspond, respectively,
to indices of refraction which are numerically equal and opposite in sign. In
general these two nappes are portions of the same inseparable analytic surface.
258. Complexes. A line complex consists of all the lines of a three-parameter
family. Let the equations of a line be given in the form
(64) x — az + p, y = bz + q .
Any line complex may be defined by means of a relation between a, b, p, q of
the form
(65) F(a,b,p,q) = 0,
and conversely. If F is a polynomial in a, b, p, q, the complex is called an
algebraic complex. The lines of the complex through any point (x0, yo, Zo) form
a cone whose vertex is at that point ; its equation may be found by eliminating
a, 6, p, q between the equations (64), (65), and
(66) x0 = az0 + p , yo = bz0 + q.
Hence the equation of this cone of the complex is
/«7\ vix ~ x° y ~ y° x°z ~ xz° Z/oZ - yz
(yi) r I ) > )
\z — ZD z - ZQ z — z0 z — Z0
Similarly, there are in any plane in space an infinite number of lines of the
complex ; these lines envelop a curve which is called a curve of the complex.
If the complex is algebraic, the order of the cone of the complex is the same as the
XII, § -5,s] FAMILIES OF STRAIGHT LINES 537
class of the curve of the complex. For, if we wish to find the number of lines of
the complex which pass through any given point A and which lie in a plane P
through that point, we may either count the number of generators in which P
cuts the cone of the complex whose vertex is at A, or we may count the number
of tangents which can be drawn from A to the curve of the complex which lies
in the plane P. As the number must be the same in either case, the theorem is
proved.
If the cone of the complex is always a plane, the complex is said to be linear,
and the equation (65) is of the form
(68) Aa + Bb + Cp + Dq + E(aq - bp) + F = 0.
Then the locus of all the lines of the complex through any given point (XQ, 3/0, ZQ)
is the plane whose equation is
( A(x - XQ) + B(y - y0) + C(x0z - z0x)
I + D(y0z - z02/) + E(yQx - x0y) + F(z - z0) = 0.
The curve of the complex, since it must be of class unity, degenerates into a
point, that is, all the lines of the complex which lie in a plane pass through a
single point of that plane, which is called the pole or the focus. A linear com
plex therefore establishes a correspondence between the points and the planes
of space, such that any point in space corresponds to a plane through that point,
and any plane to a point in that plane. A correspondence is also established
among the straight lines in space. Let D be a straight line which does not
belong to the complex, F and F' the foci of any two planes through D, and A
the line FF'. Every plane through A has its focus at its point of intersection <p
with the line D, since each of the lines </>F and <f>F' evidently belongs to the
complex. It follows that every line which meets both D and A belongs to the
complex, and, finally, that the focus of any plane through I) is the point where
that plane meets A. The lines D and A are called conjugate lines; each of them
is the locus of the foci of all planes through the other.
If the line D recedes to infinity, the planes through it become parallel, and
it is clear that the foci of a set of parallel planes lie on a straight line. There
always exists a plane such that the locus of the foci of the planes parallel to it
is perpendicular to that plane. If this particular line be taken as the z axis,
the plane whose focus is any point on the z axis is parallel to the xy plane. By
(69) the necessary and sufficient condition that this should be the case is that
A = JB = C = D = 0, and the equation of the complex takes the simple form
(70) aq-bp + K=0.
The plane whose focus is at the point (x, y, z) is given by the equation
(71) Xy -Yx + K(Z-z) = Q,
where JT, Y, Z are the running coordinates.
As an example let us determine the curves whose tangents belong to the
preceding complex. Given such a curve, whose coordinates x, y, z are known
functions of a variable parameter, the equations of the tangent at any point are
X - x _ Y-y _ Z-z
dx dy dz
538 SURFACES [XII, Exs.
The necessary and sufficient condition that this line should belong to the given
complex is that it should lie in the plane (71) whose focus is the point (x, y, z),
that is, that we should have
(72) xdy — ydx = Kdz.
We saw in § 218 how to find all possible sets of functions x, y, z of a, single
parameter which satisfy such a relation ; hence we are in a position to find
the required curves.
The results of § 218 may be stated in the language of line complexes. For
example, differentiating the equation (72) we find
(73) xd*y -yd*x = Kd*z,
and the equations (72) and (73) show that the osculating plane at the point
(x, y, z) is precisely the tangent plane (71); hence we may state the following
theorem :
// all the tangents to a skew curve belong to a linear line complex, the osculating
plane at any point of that curve is the plane whose focus is at that point.
(APPELL.)
Suppose that we wished to draw the osculating planes from any point 0 in
space to a skew curve F whose tangents all belong to a linear line complex. Let
M be the point of contact of one of these planes. By Appell's theorem, the
straight line M 0 belongs to the complex ; hence M lies in the plane whose focus
is the point 0. Conversely, if the point M of T lies in that plane, the straight
line M O, which belongs to the complex, lies in the osculating plane at M ; hence
that osculating plane passes through O. It follows that the required points are
the intersections of the curve with the plane whose focus is the point 0 (see
§ 218).
Linear line complexes occur in many geometrical and mechanical applica
tions. The reader is referred, for example, to the theses of Appell and Picard.*
EXERCISES
1. Find the lines of curvature of the developable surface which is the
envelope of the family of planes defined in rectangular coordinates by the
equation
z = ax + y<f>(a) + R Vl + a2 + <p'2(a) ,
where a is a variable parameter, <f>(a) an arbitrary function of that parameter,
and R a given constant.
[Licence, Paris, August, 1871.]
2. Find the conditions that the lines x = az + a, y - bz + /3, where a, 6, a, /3
are functions of a variable parameter, should form a developable surface for
which all of the system of lines of curvature perpendicular to the generators lie
on a system of concentric spheres.
[Licence, Paris, July, 1872.]
* Annales scientifiques tie I'Ecole Normale superieure, 1876 and 1877.
XII, ExS.] EXERCISES 539
3. Determine the lines of curvature of the surface whose equation in rec
tangular coordinates is
ez — cos x cos y .
[Licence, Paris, July, 1875.]
4. Consider the ellipsoid of three unequal axes denned by the equation
x2 y2 z2
* + * + *-1 = 0-
and the elliptical section E in the xz plane. Find, at each point M of E : 1) the
values of the principal radii of curvature B\ and -R2 of the ellipsoid, 2) the rela
tion between HI and R%, 3) the loci of the centers of curvature of the principal
sections as the point M describes the ellipse E.
[Licence, Paris, November, 1877.]
5. Derive the equation of the second degree for the principal radii of curva
ture at any point of the paraboloid defined by the equation
/j.2 ift
- + T = 2z'
a 6
Also express, in terms of the variable z, each of the principal radii of curva
ture at any point on the line of intersection of the preceding paraboloid and the
paraboloid denned by the equation
a - X b - X
[Licence, Paris, November, 1880.]
6. Find the loci of the centers of curvature of the principal sections of the
paraboloid defined by the equation xy = az as the point of the surface describes
the x axis.
[Licence, Paris, July, 1883.]
7. Find the equation of the surface which is the locus of the centers of cur
vature of all the plane sections of a given surface S by planes which all pass
through the same point M of the surface.
8. Let M T be any tangent line at a point If of a given quadric surface, 0 the
center of curvature of the section of the surface by any plane through MT,
and 0' the center of curvature of the evolute of that plane section. Find the
locus of 0' as the secant plane revolves about MT.
[Licence, Clermont, July, 1883.]
9. Find the asymptotic lines on the anchor ring formed by revolving a circle
about one of its tangents.
[Licence, Paris, November, 1882.]
10. Let C be a given curve in the xz plane in a system of rectangular coordi
nates. A surface is described by a circle whose plane remains parallel to the
xy plane and whose center describes the curve C, while the radius varies in such
a way that the circle always meets the z axis. Derive the differential equation
of the asymptotic lines on this surface, taking as the variable parameters the
540 SURFACES [xn, Exs.
coordinate z of any point, and the angle 0 which the radius of the circle through
the point makes with the trace of the plane of the circle on the xz plane.
Apply the result to the particular case where the curve C is a parabola
whose vertex is at the origin and whose axis is the x axis.
[Licence, Paris, July, 1880.]
11. Determine the asymptotic lines on a ruled surface which is tangent to
another ruled surface at every point of a generator A of the second surface,
every generator of the first surface meeting A at some point.
12. Determine the curves on a rectilinear helicoid whose osculating plane
always contains the normal to the surface.
[Licence, Paris, July, 1876.]
13. Find the asymptotic lines on the ruled surface defined by the equations
x = (1 + u) cosv, y — (\ — w)sinv, z = u.
[Licence, Nancy, November, 1900.]
14*. The sections of a surface S by planes through a straight line A and the
curves of contact of the cones circumscribed about S with their vertices on A
form a conjugate network on the surface.
[KOENIGS.]
15*. As a rigid straight line moves in such a way that three fixed points
upon it always remain in three mutually perpendicular planes, the straight line
always remains normal to a family of parallel surfaces. One of the family of
surfaces is the locus of the middle point of the segment of the given line bounded
by the point where the line meets one of the coordinate planes and by the foot
of the perpendicular let fall upon the line from the origin of coordinates.
[DARBOUX, Comptes rendus, Vol. XCII, p. 446, 1881.]
16*. On any surface one imaginary line of curvature is the locus of the points
for which the equation 1 + p2 + q2 — 0 is satisfied.
[In order to prove this, put the differential equation of the lines of curvature1
in the form
(dp dy - dq dx)(l + p* + q-) + (p dy - q dx)(p dp + q dq) = 0.]
[DARBOUX, Annales de VEcole normals, 1864.]
INDEX
[Titles in italic are proper names ; numbers in italic are page numbers ; and num
bers in roman type are paragraph numbers, which are the same as in the original
edition.]
Abdank-Abakanowicz : 201, ftn.
Abel: 153, 75; 215, 105; 348, 166;
577, 177.
Abel's lemma: 153, 75 ; 378, 166 ; 379,
178; test: 348, 166; theorem: 577,
177.
Abelian integrals : see Integrals.
Absolute value : see Value.
Algebraic curves : see Curves and
Functions, implicit.
Algebraic equations : see D'Alembert's
theorem.
Ampere: 68, ftn.; 78, 42.
Ampere's transformation : 78, 42.
Amsler: 201, 102.
Amsler's planimeter : 201, 102.
Analytic extension : 385, 180.
Analytic functions, curves, etc.: see
Functions, Curves, etc.
Anomaly, eccentric : 248, ex. 19 ; 406,
189.
Apsidal surfaces : see Surfaces.
Appell : 538, 258.
Appell's theorem : 538, 258.
Approximate evaluation : see Evalua
tion.
Archimedes : 134, 64.
Arcsine, series for : 383, 179.
Arctangent, series for: 382, 179.
Area, of a curve : 135, 64 ; 15S, 76 ;
see also Quadrature ; of a closed
curve: 187, 94; see also Integral
P dx + Q dy ; of a surface : 264, ;
272, 131 ; in oblique coordinates :
191, 95 ; in polar coordinates : 189,
95.
Arndt: 356, 169.
Array : 353, 169 ; see also Double series
and Infinite series.
Assemblages : 140, 68.
Asymptotic lines : 506, 242.
Asymptotic value of F : 291, 141.
Average value theorem : see Law of
the mean.
Balitrand: 495, ex. 11.
Beltrami: 87, ex. 21.
Bernoulli, D.: 411, 195.
Bertrand : 63, 32 ; 80, ftn. ; 133, ex. 10 ;
201, 101 ; 484, 232 ; 485, 233.
Bertrand's curves : 485, 233.
Bilinear covariants : see Covariants.
Binomial differentials : 247, ex. 8.
Binomial theorem : 104, 50 ; 383, 179 ;
391, 182 ; 474, 228.
Bonnet : 4U, 196.
Bouquet : 480, 230.
Borda's series : 133, ex. 11.
Bruno, Faa de : 34, ex. 19.
Calculus of variations : see Variations.
Cardioid, length : 154, 80.
Catalan : 262, ftn. ; 294, ex. 9.
Catenary: 220, 107; 292, ex. 1; 440,
208.
Cauchy: 7, 6 ; 29, 18 ; 91, 44 ; 106, 50 ;
183, 91; 288, 140; 328, 156; 331,
157 ; 332, 159 ; 335, 160 ; 347, 165 ;
352, 168; 359, 172; 378, ftn.; 391,
182 ; 400, 187; 495, ex. 9 ; 535, 257.
Cauchy's test (series constant terms):
332, 159; theorem (series constant
terms) : 335, 161 ; theorem (integral
convergence test) : 359, 172.
Caustics : 432, 204 ; 536, 257.
Cayley: 279, 134.
Change of variables : see also Transfor
mation ; definite integrals: 166, 84;
541
542
INDEX
double integrals: 264, 127; line in
tegrals : 186, 93 ; multiple integrals :
312, 150 ; triple integrals : 300, 145.
Characteristic curve: 459, 219.
Chasles : 166, 83.
Chasles' theorem : 166, 83.
Center of curvature : see Curvature.
Cesaro: 495, exs. 10 and 11.
Complexes (algebraic, linear, cone of,
etc.): 536, 258.
Complex variable : 575.
Conditional convergence : see Con
vergence.
Congruences, line : 53.7,255; linear:
533, 255 ; of normals : 533, 256.
Conies (as unicursal curves) : 21 5, 105 ;
222, 108.
Conjugate curves: 495, ex. 8; lines:
511, 245 ; tangents : see Tangents.
Conoid : 509, 243.
Contact (of plane curves): 443, 211;
(of skew curves) : 4S6, 234 ; (of
curves and surfaces) : 490, 236.
Contact transformations : see Trans
formations.
Continuity (definition) : 2, 3 ; 12, 10 ;
uniform : 144, 70 ; 251, 120.
Continuous functions : see Functions.
Contour lines : 262, 125.
Convergence : 327, 156 ; 350, 167; 354,
169; 359, 171; see also Infinite series,
Integrals, etc. ; absolute : 344, 164 ;
351, 167 ; 555, 169; conditional: 347,
165; interval of: 375, 177; of inte
grals: 369, 175; uniform: 367,174.
Convex surfaces : 500, 239.
Coordinates, elliptic: 268, 129; 307,
147 ; orthogonal : see Orthogonal
systems ; polar : 31, ex. 1 ; 66, 34 ;
74, 38 ; 268, 129.
Corner (of a curve) : 6, 5.
Cotes: 199, 100.
Covariants, bilinear: 87, ex. 20.
Cubic curves (unicursal) : 222, 108.
Cubic forms : see Forms.
Curvature, of a plane curve (center of) :
63, 32 ; 433, 205 ; (circle of) : 63, 32 ;
434, 205 ; 448, 213 ; (radius of) : 63,
32; 66, 34; 433, 205; of a skew
curve (center, radius, etc.): 469, 225;
471, 226 ; of a surface : 497, 239 ff. ;
(principal centers of) : 128, 61 ; 501,
240; (principal radii of): 128, 61;
501, 240; 503, 241; 519, 249; lines
of : 514, 246 ; 521, 250, 251.
Curves, algebraic : 221, 108 ; see also
Functions, implicit ; analytic : 407,
192 ; 409, 193 ; deficiency of : 221,
108 ; see also Curves, unicursal ;
plane : 5, 5 ; 62, 32 ; 92, 45 ; 407,
192 ; 426, 201 ff . ; regular : 408, 192 ;
skew: 5, 5; 51, 27; 409, 193; 453,
215 ff.; unicursal: 215, 105; 221,
108.
Curvilinear integrals : see Integrals,
line.
Cusp : 113, 53 ; 403, 192.
Cuspidal edge: see Striction, line of.
Cuts (for periodic function) : 3 IS, 153.
Cyclide (Dupin's) : 524, 252.
Cycloid : 438, 207.
D'Alentbert: 131, 63; 201, 142; 332,
159.
D'Alembert's test (series of constant
terms) : 332, 159 ; theorem : 131,
63 ; 291, 142 ; see also Roots, exist
ence of.
Darboux: 6, ftn.; 140, ftn. ; 151, 73;
524, 251; 540, exs. 15 and 16.
Darboux's theorem : 151, 73.
Deficiency : 221, 108 ; see also Inte
grals, Abelian, and Curves, unicursal.
Definite integrals : see Integrals.
De V Hospital: see L"1 Hospital, de.
Density : 296, 143.
Derivatives, definition of : 5, 5 ; 12,11;
J7, 13; extensions of definition of:
17, 13 ; 265, 127 ; of implicit func
tions : 38, 21 ; 40, 23 ; 42, 24 ; rules
for: 15, 11.
Descartes' folium : 246, ex. 2.
Determinants, functional : see Func
tional determinants.
Developable surfaces : see Surfaces.
Development in series : 405, 189 ; see
also Infinite series, Taylor's series,
Fourier's series, etc.
INDEX
543
Dextrorsal (skew curve) : 476, 228.
Differential equations : see Equations ;
invariants: see Invariants; notation:
19, 14 ff. ; parameters : 81, 43 ; 87,
ex. 21 ; 510, 244.
Differentials, binomial : see Binomial
differentials ; definition of : 19, 14 ;
higher orders: 20, 14; total: S3,
16 ; 313, 151 ff. ; see also Integrals,
line, and Integral P dx + Q dy.
Differentiation, order immaterial : 13,
11; of integrals: 154, 76; 192, 97;
194, 97; 368, 175 ; 370, 175; of line
integrals: 194, 97; of series: 364,
174; 380, 179; 405, 189.
Direction cosines: 164, 81.
Direct path (for periodic functions) :
319, 153.
Dirichlet: 250, 124; 308, 148; 347,
165 ; ^, 196.
Dirichlet's integrals : 308, 148 ; condi
tions: 414, 195.
Discontinuity : 4, 4 ; 6, ftn.
Discontinuous functions: see Func
tions.
Division of series: 392, 183.
Dominant functions : see Functions.
Double integrals : see Integrals, double.
Double points, of a curve : 112, 53 ;
221, 108; of involutions: 232, 112;
of unicursal curves : 222, 108.
Double power series : see Power series,
double.
Double series: 353, 169; 367, 174;
388, 182 ; see also Infinite series
and Substitution of series.
Duhamel: 135, ftn.; 151, ftn.; 340,
163.
Duhamel's test (series of constant
terms) : 340, 163.
Dupin: 521, 251; 524, 252 ; 535, 257.
Dupin's cyclide : 524, 252 ; theorem :
521, 251.
Eccentric anomaly : see Anomaly.
Edge, cuspidal: see Striction, line of;
of regression : 463, 221.
Element of area: 267, 128 ; 275, 132.
Element of volume : 304, 146.
Ellipse, area of: 189, 94; 220, 106;
248, ex. 19; length of: 234, 112;
366, 174.
Ellipsoid, area of : 294, ex. 9 ; volume
of: 285, 137.
Elliptic coordinates : see Coordinates ;
functions : see Functions ; integrals :
see Integrals ; points : 500, 239.
Envelopes (of plane curves) : 426, 201 ;
(of skew curves) : 465, 223 ; (of sur
faces) : 459, 219.
Epicycloid : 452, ex. 7.
Equations, differential (developable
surfaces, asymptotic lines, etc.) : see
Surfaces, developable ; Asymptotic
lines, etc.
Equations, total differential : see Dif
ferentials, total.
Equations, partial differential (classi
fication) : 73, 38 ; (Laplace's) : 80,
43 ; (reduction of) : 72, 38.
Equations, intrinsic (of a curve) : 441,
210; reciprocal: 234, 113; solutions
of : see Roots and D'Alembert's
theorem ; tangential (ol a curve) :
207, ex. 21.
Error, limit of : see Remainder and
Evaluation.
Euleri 184, 92; 236, 113; 246, ex. 4;
280, 134 ; 411, 195 ; 501, 240.
Euler's constant : 103, 49 ; integrals :
184, 92 ; 280, 134 ; see also Function
T ; theorem (surfaces) : 501, 240.
Evaluation, approximate (of definite
integrals) : 197, 99 ; 199, 100 ; 201,
101 ; 207, ex. 24 ; see also Pla-
nimeter ; (factorials) : 291, 141 ;
(log T) : 291, 141.
Evaluation of integrals : 287, 140 ;
254, 123 ; 297, 143 ; 811, 160 ; 373,
176.
Evolute: 432, 204; 436, 206; 480,
231 ; 516, 247.
Existence of roots : see Roots and
Functions, implicit.
Exponential, series for : 100, 48.
Extrema: 116, 65; 118, 56; 125, 69;
128, 61 ; 251, 120 ; see also Maxima
and Minima.
544
INDEX
Families (of straight lines) : 526,
253 ff.
Fermat : 137, 66.
Finite functions : see Functions.
Focal planes : 532, 255 ; points : 531,
255 ; surface : 531, 255.
Forms, binary cubic : 60, 30 ; in
determinate : see Indeterminate
forms.
Formulse of reduction : see Reduction
formulae.
Fourier: 418, 197.
Fourier's series: 418, 197 ff. ; see also
Trigonometric series.
Franklin: 257, 123.
Frenet : 477, 229.
Frenet's formulae : 477, 229.
Fresnel: 86, ex. 17.
Fresnel's wave surface: 86, ex. 17.
Functional determinants: 46, 25; 52,
28 ; 58, 29 ; 265, 127 ; 304, 146.
Functions, analytic : 407, 191 ff. ;
B (p, q) : 279, 134 ; continuous : 143,
70; 200, ftn.; 250, 120; 362, 173;
378, 178; 422, 199; see also Con
tinuity ; discontinuous : ^, 4 ; 161,
79; 195, 98; 205, exs. 6 and 7;
denned by integrals : 192, 97 ; 195,
98; 221, 108; dominant: 386, 181;
^ 396, 186; elliptic: £33, 112; expo
nential : see Exponential ; T(a) : 183,
92 ; 279, 134 ; 290, 141 ; 308, 148 ;
313, 150; homogeneous: £9, 18;
hyperbolic: 219, 106; implicit: 35,
20; -45, 25; 395, 187; integrable :
147, 72 ; 205, ex. 8 ; inverse : 41,
23 ; 50, 26 ; ^06, 190 ; logarithmic :
see Logarithm ; monotonic : 148,
72; periodic: SIS, 153; primitive:
see Primitive functions and Inte
grals; rational: 3, 3; 158, 77; 205,
ex. 12 ; £05, 103; real variables: 2,
2; 11, 10; (etc., see special titles) ;
transcendental : 221, 108 ; 236, 114 ;
422, 199; trigonometric: 700, 48;
220, 106; £36, 114; trigonometric
(inverse): 104, 50; 35£, 179.
Fundamental theorem of Algebra : see
D'Alembert's theorem.
Galileo : 439, 207.
Gamma function: see Function T(a).
Gauss: 199, 101; 291, 142; 344, 163.
Geometry of higher dimensions : see
Higher dimensions.
Gergonne: 535, 257.
Goursat: 35, ftn.; 45, ftn.; 88, ex. 23.
Graves : 166, 83.
Graves' theorem : 166, 83.
Greatest limit : 328, 156 ; 335, 160 ;
351, 167 ; 377, 177.
Green : 262, 126 ; 288, 140 ; 309, 149 ;
316, 152 ; 318, 153.
Green's theorem : 288, 140 ; 305, 149 ;
316, 152 ; 375, 153.
Hadamard: 378, ftn.
Halphen: 33, ex. 11; 56, ex. 18.
Harmonic series : 103, 49 ; 347, 165.
Haro's series: 183, ex. 11.
Helicoid : 509, 243 ; 579, 249.
Helix: 482, 231; ^53, 232.
Hermite : 97, 46 ; 777, 87; 205, ex. 12 ;
236, ftn. ^-j.,,
Hessians : 55, 30.
Higher dimensions : 370, 150.
Highest common divisor : £77, 104.
Hilbert: 171, 87.
Hospital, de V : see L"1 Hospital, de.
Houel: 219, 106.
Hyperbola, area of : 218, 106.
Hyperbolic functions : see Functions.
Hyperbolic point : 500, 239.
Hypocycloid: 242, 117; ^5£, ex. 7.
Implicit functions : see Functions and
D'Alembert's theorem.
Improper integrals : see Integrals.
Incommensurable numbers : 7^£, ftn. ;
777, 87; 249, ex. 21.
Indefinite integrals : see Integrals.
Independence of Path: 376, 152; see
also Integral P dx + Qdy; of sur~
face ; 3£3, 155 ; see also Integrals,
surface.
Indeterminate forms : 70, 8 ; 97, 47.
Index (of a function): 757, 77; £05,
exs. 11 and 12 ; 322, 154.
Indicatrix: 507, 240 ff.
INDEX
545
Infinite, definition of : 4, 4.
Infinite limits : see Integrals, improper.
Infinite series: 2, 1; 99, ftn.; 183,
91; 327, 156; alternating: 182,
91 ; complex terms : 350, 167 ff. ;
constant terms: 329, 157 ff.; devel
opment in : 93, 46 ; 98, 48 ; 201, 101 ;
375, 177; 404, 189; 411, 195; 418,
197; 422, 199; see also Taylor's
series, etc. ; differentiation of : 364,
174 ; 380, 179; 405, 189 ; division of :
392, 183 ; dominant : see Functions,
dominant ; Fourier's : 418, 197 ff . ;
harmonic : 103, 49 ; 347, 165 ; of in
finite series : see Substitution of series
and Double series; integration of:
201, 101; 364, 174; 368, 174; Mc-
Laurin's : see McLaurin's series ;
multiplication of: 331, 158; 379,
178; positive terms: 2, 1; 329,
157 ff. ; reversion of : 407, 100 ;
substitution of : see Substitution of
series; sum of: 99, ftn.; 329, 157;
see also Convergence ; Taylor's : see
Taylor's series ; trigonometric : 4H,
195 ff.; variable terms: 860, 173;
see also Double series, etc.
Infinitely small quantity : 19, 14 ; see
also Infinitesimal.
Infinitesimal: 19, 14; 150, 72; 252,
120.
Integrable functions : see Functions.
Integrals, Abelian : 215, 105; 221, 108;
226, 110 ; differentiation of : see Dif
ferentiation of integrals ; definite :
149, 68 ff. ; see also Evaluation of
integrals ; double : 250, 120 ff.; ellip
tic : 226, 110; 231, 112; 233, 112;
246, ex. 6 ; functions defined by :
see Functions ; hyperelliptic : 226,
110; improper: 175, 89; 179, 90;
183, 91 ; 236, 140 ; 277, 133 ; 289,
161; 553,173; 369, lib; indefinite:
154, 76 ; 208, 103 ff. ; see also Func
tions, primitive, and Evaluation of
integrals; integration of: see Inte
gration of integrals ; line : 184, 93 ;
201, 102 ; 263, 126 ; 316, 152 ; 322,
155 ; see also Differentials, total, and
Integral Pdx + Qdy, and Green's
theorem; logarithm: 245, 118 ; mul
tiple : 310, 150 ff. ; 358, 171 ; 367,
174 ; P dx + Q dy : 316, 152 ff. ;
pseudo-elliptic : 234, 113 ; 246, ex. 5 ;
247, ex. 7 ; surface : 280, 135 ff. ; 322,
155; triple: 296, 143; xdy -ydx:
189, 94 ; 191, 96 ; 206, ex. 14.
Integraphs: 201, 102.
Integration, of binomial differentials :
224, 109; of integrals: 256, 123;
mechanical : 201, 102 ; of series :
201, 101 ; 364, 174 ; 368, 174 ; see
also Integrals.
Interpolation: 198, 100.
Interval : function defined in : 2, 2 ; 7,
5 ; of convergence : see Convergence.
Intrinsic equations : see Equations.
Invariants : 59, 30 ; 70, 37.
Inverse functions : see Functions.
Inversion, of functions : see Func
tions ; transformation of : 66, 35 ;
69, 36.
Involutes: ^5^,204; 436,206; 480, 231.
Involutions: 231, 112; 234, 113; 247,
ex. 7.
Involutory transformations : 69, 36 ;
78, 41 ; 79, 42.
Jacobi: 22, ftn.; 3#, ex. 5; 46, ftn.;
55, ftn.
Jacobians : see Functional determi
nants.
Jamet : 509, 243.
Joachimsthal : 520, 250.
Joachimsthal's theorem : 520, 250.
Jordan : 360, ftn.
Kelvin, Lord : 55, ex. 10.
Kepler: 406, 189.
Kepler's equation : 249, ex. 19 ; 406,
189.
Koenigs : 540, ex. 14.
Lagrange : 5, 5 ; 7, 6 ; 20, 18 ; 90, 44 ;
198, 100 ; 274, 131 ; ^-4, 189 ; 445,
212.
Lagrange's formula (implicit func
tions) : 34, ex. 8 ; 404, 189 ; formula
546
INDEX
(interpolation) : 198, 100 ; identity :
274, 131.
Lame: 80, ftn.; 82, 43; 325, ex. 10.
Laplace : 73, 38 ; 84, ex. 8 ; 404, 189.
Laplace's equation : 73, 38.
Laugel: 140, ftn.
Law of the mean, for derivatives : 8,
8; 25,11; 16, ftn.; 55,48; 255,76;
£05, 127; for integrals (1st law) : 252,
74 ; 253, 121 ; for integrals (2d law) :
252, 74; 205, ex.13; generalizations:
10, 8 ; 98, 48 ; 265, 127.
Lebesgue: 422, 199.
Legendre: 33, ex. 9; 6<9, 36; 275, 88;
366, 174; 394, 184.
Legendre's polynomials : 33, ex. 9 ;
275, 88; 201, 101; formula: ^252,
203; integrals: 233, 112; 566, 174;
394, 184 ; transformation : 68, 36 ;
77, 41.
Leibniz : 7,6; 19, ; 27, 17 ; 29, 18.
Leibniz' formula : 27, 17.
Lemniscate: 22S, 108; £5^, 112.
Length : 262, 80 ; 26^, 80 ; etc.
Lie, : 68, ftn.
L"1 Hospital, dei 10, 8.
L' Hospital, de, theorem: 10, 8.
Limit : 2, 1 ; a lower : 140, 68 ; an
upper: 91, ftn.; 140, 62; greatest:
see Greatest limit ; of error : see
Evaluation ; of integration : see In
tegrals; the lower: 142, 68; the
upper : 2^2, 62.
Line complexes : see Complexes ; con
gruences : see Congruences ; inte
grals : see Integrals.
Line of curvature : see Curvature.
Line of striction : see Striction.
Linear transformations : see Trans
formations.
Liouville: 231, 111.
Logarithm : .57, 28 ; 100, 49 ; 102, 49;
382, 179.
Loop-circuit: 523, 153.
Lyon : 4$4, ftn.
Mains : 555, 257.
Mams' theorem : 555, 257.
Mannheim : 495, exs. 7 and 11 ; 524, 252.
Mansion : 207, ex. 24.
Mass: 296, 143.
Maximum : 3,3; 116, 55 ff. ; £52, 120 ;
see also Extremum.
McLaurin's series: 99, 48; 382, 179;
see also Taylor's series.
Mean, law of the : see Law.
Mechanical quadrature : 201, 102.
Mertens: 352, 168.
Meusnier: 497, 239.
Meusnier's theorem : 497, 239.
Minimum : 5, 3 ; 226, 55 ; see also
Extremum.
Mobius' strip : 280, ftn.
Monge: 29, 18 ; 44, 24 ; 523, 251.
Monotonically increasing functions :
see Functions, monotonic.
Multiple series: 310, 150 ff., 358, 171;
567, 174.
Multiplication of series : see Series.
Murphy: 373, ex. 1.
Newton : 19, ftn.
Normal sections : 497, 239 ; 501, 240.
Normals, congruence of : 555, 256 ;
length of : 30, 19 ; plane curves : 30,
19; principal (skew curves): 471,
226.
Numbers, incommensurable : 142, ftn. ;
272, 87; 249, ex. 21; transcenden
tal: 272, 87.
Order of contact : see Contact,
Ordinary points : see Points.
Orthogonal systems, of curves: #75,
132; triple: 80, 43 ; 521, 251.
Oscillation : 142, 69; £52, 120; 448, 213.
Osculating plane: 453, 215 ; 455, 216;
sphere : 492, 237.
Osculation : ,95, 45 ; 448, 213 ; 453,
215 ; ^55, 216 ; 488, 235 ; 492, 237.
Osgood: 53, ftn.; 252, ftn.; 369, ftn.
Ostrogradsky : 309, ftn.
Ostrogradsky's theorem : 309, 149.
Painleve : 88, ex. 23.
Parabola : 255, 64 ; 257, 66 ; 220, 107.
Parabolic point : 500, 239 ; 520, 249.
Paraboloid: 525, 246; 523, 251.
INDEX
547
Parallel curves : 207, ex. 20 ; surfaces :
86, ex. 16.
Parameter of distribution : 528, 253 ;
530, 254.
Parameters, differential ; see Differ
ential.
Partial differential equations : see
Equations, partial differential.
Peano : 456, ftn.
Pedal curves: 69, 36; 207, exa. 21
and 22.
Pellet: 495, ex. 11.
Periodic functions : see Functions.
Periods: 318, 153.
Picard : 322, 154 ; 538, 258.
Planimeter: 201, 102.
Poincare: 386, 181.
Point transformations : see Transfor
mations.
Points, ordinary : 110, 53 ; 408, 192.
Points, singular : 110, 53 ; 114, 54 ;
319, 153 ; 408, 192 ; 409, ftn.
Poisson : 204, ex. 6 ; 325, ex. 8.
Polar coordinates : see Coordinates.
Polar line : 473, 227.
Polar surface : 473, 227.
Polynomials, continuity of : 3, 3 ; rela
tively prime : 211, 104 ; 214, 104.
Potential equation : see Laplace's equa
tion.
Power series: 375, 177 ff. ; double:
394, 186.
Primitive functions : 139, 67; 154, 76 ;
see also Integrals.
Principal normals, tangents, etc. : see
Normals, Tangents, etc.
Pringsheim: 340, 162.
Prismoid : 285, 138 ; 310, 150.
Prismoidal formula: 285, 138.
Protective transformations : see Trans
formations.
Pseudo-elliptic integrals : see Integrals.
Puiseux : 484, 232.
Quadrature : 134, 64 ; 135, 65 ; 160,
78 ; see also Area, Integrals, etc.
Quadrics, confocal : 533, 251.
Quartic curves : 223, 108.
Quetelet : 535, 257.
Raabe : 340, 163.
Raabe's test : 340, 163.
Radius of curvature, of torsion: see
Curvature, Torsion.
Rational functions : see Functions.
Reciprocal equations : see Equations ;
polars : see Transformations ; radii :
see Transformations.
Rectification of curves : see Length.
Reduction formulse : 208, 103; 210,
104; £26, 110; 227, 110; 239, 115;
240, 116; 244, 118; 248, exs. 15,
16, and 17; £49, ex. 21.
Regression, edge of: 463, 221.
Regular curves : see Curves.
Remainder (Taylor's series) : 90, 44 ;
98, 48.
Reversion of series : 407, 190.
Riccati equations : 511, 244.
Riemann: 140, ftn.; 309, ftn.; 347,
165.
Riemann's theorem : 309, ftn. ; see also
Green's theorem.
Roberts : 294, ex. 10.
Rodriguez: 33, ex. 8; 517, 248.
Rodrigues' formula : 517, 248.
Rolle's theorem : 7, 7.
Roots, existence of: 3, 3; 291, 142;
321, 154 ; see also Functions, im
plicit, and D'Alembert's theorem.
Roulette : 207, ex. 23 ; 220, 107 ; 526,
252.
Rouquet : 495, ex. 4.
Ruled surfaces : see Surfaces.
Kcheffer: 125, 56.
Schell : 495, ex. 7.
Schwarz : 11, 9.
Schwarzian : 88, ex. 22.
Sequences: 327, 156 ; see also Infinite
series.
Series : see Infinite series, Taylor's
series, Double series, etc.
Serret : 234, ftn. ; 495, ex. 7.
Serret's curves : 234, ftn.
Simpson: 199, 100.
Singular points: 110, 53; 114, 54;
319, 153; 408, 192; 409, ftn.
Sinistrorsal (skew curve) : 476, 228.
548
INDEX
Skew curves : see Curves.
Steiner : 207, ex. 23.
Stokes: 282, 136.
Stokes' theorem : 282, 136.
Striction, line of : 529, 253.
Sturm : 174, 88.
Sturin sequences : 174, 88.
Subnormal : 30, 19.
Substitution of series : 388, 182 ; 397,
186 ; see also Double series.
Substitutions : see Transformations.
Subtangent: 30, 19.
Surface integrals : see Integrals.
Surfaces: 75, 39; 497, 239 ff. ; ana
lytic: 410, 194ff. ; apsidal: 86, ex.
17; developable: 79,42; 461,221;
464, 222 ; 505, 241 ; focal : 531, 255 ;
parallel: 86, ex. 16; ruled : 285, 138 ;
509,244; 526, 253; translation : 513,
245 ; tubular : 524, 252 ; unilateral :
280, 135; wave: 86, ex. 17.
Tangential equations : 207, ex. 21.
Tangents, asymptotic : 503, 240 ; con
jugate : 511, 245 ; length of : 30, 19 ;
principal: 503, 240; stationary: 457,
217 ; to curves (plane) : 5, 5 ; 63, 32 ;
92, 45; 97,47; to curves (skew) : 5,
5 ; 51, 27 ; to surfaces : 16, 12 ; 39,
22 ; 76, 39 and ftn.
Tannery : 555, ftn.
Taylor's series : 89, 44 ff. ; 95, 48 ff. ;
171, 86 ; ^97, 51 ; 384, 180 ; 596, 185.
Tchebycheff: 257, 123.
Term-by-term differentiation : see Dif
ferentiation of series ; integration :
see Integration of series.
Tests for convergence : see Conver
gence.
Thompson, Sir Wm. : see Kelvin, Lord.
Tissot : 495, ex. 6.
Torsion and Radius of torsion : 473
and 474, 228.
Total differentials : see Differentials.
Tractrix : 441, 209.
Transcendental numbers : 171, 87.
Transformations, contact : 67, 36 ; 77,
41 ; 78, 42 ; involutory : 69, 36 ; 78,
41; 79, 42; linear : 59, 30 ; of coordi
nates: 65, 34; 76, 40; etc.; of curves:
66, 35; of independent variable : 61,
31 ; 70, 38 ; 74, 39 ; of integrals : see
Change of variables ; point : 66, 35 ;
68, 36 ; 77, 40 ; projective : 66, 35 ;
69, 37; reciprocal polars: 69, 36;
78, 41 ; reciprocal radii : 66, 35 ; 69,
36.
Trigonometric functions: see Func
tions ; series : 411, 195.
Triple integrals : see Integrals.
Triply orthogonal systems : see Orthog
onal systems.
Umbilics : 505, 241 ; 52 0, 249.
Uniform curves, continuity, conver
gence, infinitesimal : see Curves,
Continuity, Convergence, Infinitesi
mal, etc.
Unilateral surfaces : see Surfaces and
Mobius' strip.
Upper limit : see Limit.
Value, absolute : 3, 3 ; 375.
Variable, complex : 575.
Variations, calculus of : #57, 123.
Viviani: 286, 139.
Viviani's formula : 286, 139.
Volume: 254, 122 ; 284, 137; 325, ex. 8;
326, ex. 13.
Wallis: 240, 116.
Wallis' formula: 240, 116.
Wave surface : 86, ex. 17.
Weierstrass : 6, 5; 153, 75; 200, ftn.;
402, 87; 422, 199.
Weierstrass' theorem: 422, 199.
Ziwet : 406, ftn.
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